Counting Principles - IB Math AA Question Bank

Question 1

HL • Paper 1

Easy

Non Calculator

A code consists of three distinct letters chosen from $8$ available letters, followed by two distinct digits chosen from the digits $0,1,2,3,4$ .

A

Find the number of possible codes.

[2]

Write your answer here...

B

Find the number of possible codes which begin with one of three specified letters and end with an even digit.

[3]

Write your answer here...

0

Question 2

HL • Paper 1

Easy

Non Calculator

For a set of $n$ distinct objects, where $n>3$ , the number of ordered selections of three objects is ten times the number of unordered selections of two objects.

A

Write this information as an equation involving $n$ .

[2]

Write your answer here...

B

Find the value of $n$ .

[2]

Write your answer here...

0

Question 3

HL • Paper 2

Easy

Calculator Permitted

In a race there are $18$ runners. Six runners are from school A and the remaining $12$ runners are from other schools. Gold, silver and bronze medals are awarded to three different runners.

A

Find the number of possible medal outcomes.

[1]

Write your answer here...

B

Find the number of possible medal outcomes in which at least one runner from school A receives a medal.

[2]

Write your answer here...

C

Find the number of possible medal outcomes in which exactly two runners from school A receive medals.

[3]

Write your answer here...

0

Question 4

HL • Paper 2

Easy

Calculator Permitted

A file code consists of four letters followed by three digits. The letters are chosen from $A,B,C,D,E,F,G,H$ and no letter may be repeated. The digits are chosen from $0,1,2,\ldots,9$ and digits may be repeated.

A

Calculate the number of possible file codes.

[2]

Write your answer here...

B

Calculate the number of possible file codes in which at least one of the three digits is $7$ .

[3]

Write your answer here...

0

Question 5

HL • Paper 2

Easy

Calculator Permitted

Consider the binomial expansion of $(1+3x)^{-\frac{2}{3}}$ .

A

Write down the first three non-zero terms in ascending powers of $x$ .

[3]

Write your answer here...

B

State the interval of values of $x$ for which this expansion is valid.

[1]

Write your answer here...

0

Question 6

HL • Paper 1

Medium

Non Calculator

A committee of four people is to be chosen from $6$ mathematicians and $5$ physicists.

A

Write down the total number of possible committees if there are no restrictions.

[1]

Write your answer here...

B

Find the number of committees which contain at least one physicist and at least two mathematicians.

[3]

Write your answer here...

C

For each committee counted in part (b), a chair and a secretary are chosen from the mathematicians on the committee. Find the number of possible outcomes.

[3]

Write your answer here...

0

Question 7

HL • Paper 1

Medium

Non Calculator

Seven students, $A,B,C,D,E,F$ and $G$ , are to stand in a straight line.

A

Find the number of arrangements in which $A$ and $B$ stand next to each other.

[2]

Write your answer here...

B

Find the number of arrangements in which $A$ and $B$ do not stand next to each other.

[2]

Write your answer here...

C

Find the number of arrangements in which $C$ is first and $A$ and $B$ do not stand next to each other.

[2]

Write your answer here...

0

Question 8

HL • Paper 1

Medium

Non Calculator

Consider the expansion of $(1-2x)^{-3}$ in ascending powers of $x$ .

A

Find the expansion up to and including the term in $x^3$ , and state the restriction on $x$ for which the expansion is valid.

[4]

Write your answer here...

B

Hence find the coefficient of $x^3$ in the expansion of $(1+x)(1-2x)^{-3}$ .

[2]

Write your answer here...

0

Question 9

HL • Paper 1

Medium

Non Calculator

The expression $(4+x)^{\frac{1}{2}}$ is to be expanded in ascending powers of $x$ .

A

State the restriction on $x$ for which the binomial expansion is valid.

[1]

Write your answer here...

B

Find the expansion up to and including the term in $x^3$ .

[3]

Write your answer here...

C

Use your expansion to approximate $\sqrt{5}$ as a fraction.

[2]

Write your answer here...

0

Question 10

HL • Paper 1

Medium

Non Calculator

A team of five students is to be selected from $8$ students, including Lea and Max. Two different roles, leader and treasurer, are then assigned to two members of the team.

A

Find the number of possible outcomes if there are no restrictions.

[2]

Write your answer here...

B

Find the number of possible outcomes if Lea and Max cannot both be on the team.

[2]

Write your answer here...

C

Find the number of possible outcomes if Lea is on the team and Max is the treasurer.

[2]

Write your answer here...

0

Question 11

HL • Paper 1

Medium

Non Calculator

The function $f$ is defined by
$f(x)=\frac{1}{(2-x)^2}$

A

Find the binomial expansion of $f(x)$ up to and including the term in $x^3$ .

[4]

Write your answer here...

B

State the restriction on $x$ for which the expansion is valid.

[1]

Write your answer here...

C

Use the expansion in part (a) to approximate $f\left(\frac{1}{3}\right)$ as a fraction.

[1]

Write your answer here...

0

Question 12

HL • Paper 2

Medium

Calculator Permitted

A seven-digit access code is formed using digits from $0,1,2,\ldots,9$ . No digit may be repeated, and the first digit cannot be $0$ .

A

Find the number of possible access codes.

[2]

Write your answer here...

B

Find the number of possible access codes which contain exactly two odd digits.

[4]

Write your answer here...

0

Question 13

HL • Paper 2

Medium

Calculator Permitted

A debating squad is made up of students in two year groups, as shown in the table. A team of $5$ students is to be selected.

Year group	Number of students
Year group A	9
Year group B	7

A

Determine the number of teams that contain at least two students from each year group.

[3]

Write your answer here...

B

For each such team, a captain and a deputy captain are chosen from the $5$ team members. Determine the number of possible teams with these two roles assigned.

[2]

Write your answer here...

0

Question 14

HL • Paper 2

Medium

Calculator Permitted

In a lottery, a ticket consists of $6$ different numbers chosen from the numbers $1$ to $45$ . The order in which the numbers are chosen does not matter.

A

Calculate the number of different lottery tickets possible.

[1]

Write your answer here...

B

winning ticket has been drawn. Calculate the number of tickets which match exactly four of the six winning numbers.

[2]

Write your answer here...

C

Suppose $250000$ different tickets are sold for one draw. Calculate the probability that the jackpot ticket has been sold.

[2]

Write your answer here...

0

Question 15

HL • Paper 2

Medium

Calculator Permitted

The expression $(9-2x)^{\frac{1}{2}}$ may be expanded using the extended binomial theorem.

A

Show that, up to and including the term in $x^2$ , $(9-2x)^{\frac{1}{2}}=3-\dfrac{x}{3}-\dfrac{x^2}{54}+\cdots$ .

[3]

Write your answer here...

B

Hence find an approximation to $\sqrt{8.6}$ , giving your answer to three significant figures.

[2]

Write your answer here...

0

Question 16

HL • Paper 2

Medium

Calculator Permitted

Consider $f(x)=(1-4x)^{-\frac{1}{2}}$ .

A

Find the first four non-zero terms in the binomial expansion of $f(x)$ in ascending powers of $x$ .

[3]

Write your answer here...

B

Use this expansion to estimate $\displaystyle\int_0^{0.1} f(x)\,\mathrm{d}x$ , giving your answer to four decimal places.

[2]

Write your answer here...

0

Question 17

HL • Paper 1

Medium

Non Calculator

A robot moves on a rectangular grid from $S(0,0)$ to $T(5,3)$ . At each step it moves either one unit to the right or one unit upwards. Two points on the grid are $P(2,1)$ and $Q(4,2)$ .

A rectangular lattice grid with start point S at the lower left and endpoint T at the upper right. Points P and Q are labelled on interior lattice points. The diagram shows only horizontal and vertical grid lines and does not show any path.

A

Find the total number of possible paths from $S$ to $T$ .

[1]

Write your answer here...

B

Find the number of possible paths from $S$ to $T$ which pass through $P$ .

[2]

Write your answer here...

C

Find the number of possible paths from $S$ to $T$ which pass through exactly one of $P$ and $Q$ .

[4]

Write your answer here...

0

Question 18

HL • Paper 1

Medium

Non Calculator

Let $a$ and $b$ be positive integers with $a<b$ . The expansion of
$(1+ax)^{-\frac{1}{2}}(1+bx)$
in ascending powers of $x$ begins $1+3x-4x^2+\cdots$ .

A

Find the expansion of $(1+ax)^{-\frac{1}{2}}$ up to and including the term in $x^2$ .

[2]

Write your answer here...

B

Find the value of $a$ and the value of $b$ .

[4]

Write your answer here...

0

Question 19

HL • Paper 1

Medium

Non Calculator

Consider the product
$(1+x)^{-2}(1-3x)^{\frac{1}{2}}$

A

Write the expansions of both factors up to and including the term in $x^3$ .

[3]

Write your answer here...

B

Determine the coefficient of $x^3$ in the product.

[2]

Write your answer here...

0

Question 20

HL • Paper 1

Medium

Non Calculator

The expansion of $(1+px)^{-2}$ in ascending powers of $x$ begins $1-10x+qx^2+\cdots$ , where $p$ and $q$ are constants.

A

Find the value of $p$ and the value of $q$ .

[3]

Write your answer here...

B

State the restriction on $x$ for which the expansion is valid.

[1]

Write your answer here...

C

Hence find the coefficient of $x^3$ in the expansion of $(1+x)(1+5x)^{-2}$ .

[2]

Write your answer here...

0

Question 21

HL • Paper 2

Medium

Calculator Permitted

A shelf is to contain $10$ different books. Four of the books are mathematics books and the remaining six books are from other subjects.

A

Find the number of arrangements in which the four mathematics books are together.

[2]

Write your answer here...

B

Find the number of arrangements in which no two mathematics books are next to each other.

[3]

Write your answer here...

0

Question 22

HL • Paper 2

Medium

Calculator Permitted

The binomial expansion of $(a+2x)^{-\frac{1}{2}}$ begins $\dfrac13-\dfrac{x}{27}+cx^2+\cdots$ , where $a>0$ .

A

Determine the value of $a$ .

[2]

Write your answer here...

B

Determine the value of $c$ .

[2]

Write your answer here...

C

State the interval of values of $x$ for which this expansion is valid.

[1]

Write your answer here...

0

Question 23

HL • Paper 2

Medium

Calculator Permitted

The extended binomial theorem may be used to approximate cube roots.

A

Find the first four non-zero terms in the expansion of $(1+x)^{\frac13}$ in ascending powers of $x$ .

[3]

Write your answer here...

B

Hence find an approximation to $\sqrt[3]{31}$ , giving your answer to three significant figures.

[3]

Write your answer here...

0

Question 24

HL • Paper 1

Medium

Non Calculator

Eight different paintings are to be displayed in a straight line. Three of the paintings are landscapes, two are portraits and three are abstracts.

A

I.

Write down the number of possible arrangements if there are no restrictions.

[1]

Write your answer here...

II.

Find the number of possible arrangements in which the three landscapes are displayed together.

[2]

Write your answer here...

B

Find the number of possible arrangements in which no two landscapes are next to each other.

[3]

Write your answer here...

C

Find the number of possible arrangements in which exactly two of the landscapes are next to each other.

[4]

Write your answer here...

0

Question 25

HL • Paper 1

Medium

Non Calculator

Consider the function

f(x)=(9-6x)^{-\frac12}

A

I.

Rewrite $f(x)$ in the form $a(1+bx)^{-\frac12}$ and state the restriction on $x$ for which its binomial expansion is valid.

[2]

Write your answer here...

II.

Find the binomial expansion of $f(x)$ up to and including the term in $x^3$ .

[3]

Write your answer here...

B

Use the expansion in part (a) to approximate $\dfrac{1}{\sqrt{8}}$ as a fraction.

[3]

Write your answer here...

C

Hence find the coefficient of $x^3$ in the expansion of $(1+2x)f(x)$ .

[2]

Write your answer here...

0

Question 26

HL • Paper 1

Medium

Non Calculator

The expansion of $(4+px)^{\frac32}$ in ascending powers of $x$ begins

8+12x+qx^2+rx^3+\cdots

where $p$ , $q$ and $r$ are constants.

A

I.

Show that $p=4$ .

[2]

Write your answer here...

II.

Find $q$ and $r$ .

[3]

Write your answer here...

B

State the restriction on $x$ for which the expansion with $p=4$ is valid.

[1]

Write your answer here...

C

Use the expansion up to and including the term in $x^3$ to approximate $5^{\frac32}$ as a fraction.

[3]

Write your answer here...

0

Question 27

HL • Paper 2

Medium

Calculator Permitted

The extended binomial theorem may be used to approximate $\sqrt[3]{64.8}$ .

A

I.

Find the expansion of $(1+x)^{\frac13}$ up to and including the term in $x^4$ .

[4]

Write your answer here...

II.

State the restriction on $x$ for which this expansion is valid.

[1]

Write your answer here...

B

Show that $\sqrt[3]{64.8}=4(1+0.0125)^{\frac13}$ .

[2]

Write your answer here...

C

Hence use the expansion in part (a) to approximate $\sqrt[3]{64.8}$ , giving your answer to five significant figures.

[3]

Write your answer here...

D

Explain why the binomial expansion is appropriate for this approximation.

[1]

Write your answer here...

0

Question 28

HL • Paper 2

Medium

Calculator Permitted

Let $g(x)=(1+x)^{-3}(1-2x)^{\frac{1}{2}}$ .

A

Write down the terms up to and including the term in $x^4$ for each of $(1+x)^{-3}$ and $(1-2x)^{\frac{1}{2}}$ .

[4]

Write your answer here...

B

Determine the coefficient of $x^4$ in the expansion of $g(x)$ .

[2]

Write your answer here...

0

Question 29

HL • Paper 1

Hard

Non Calculator

A symposium has $5$ analysis specialists, $4$ geometry specialists and $3$ statistics specialists. A panel of $5$ people is to be selected.

A

I.

Write down the total number of possible panels if there are no restrictions.

[1]

Write your answer here...

II.

Find the number of possible panels containing exactly two geometry specialists and at least one specialist from each of the other two areas.

[3]

Write your answer here...

B

Find the number of possible panels that contain at least one specialist from each area and more analysis specialists than statistics specialists.

[4]

Write your answer here...

C

For each panel counted in part (b), a chair and a secretary are chosen from different specialist areas. Find the number of possible outcomes.

[4]

Write your answer here...

0

Question 30

HL • Paper 1

Hard

Non Calculator

Five distinct tasks are to be assigned to workers. There are $8$ workers available, of whom $X$ , $Y$ and $Z$ are trainees. Each task is assigned to exactly one worker, and no worker may be assigned more than one task.

A

I.

Find the number of possible assignments if there are no restrictions.

[2]

Write your answer here...

II.

Find the number of possible assignments in which worker $X$ is assigned the first task.

[2]

Write your answer here...

B

Find the number of possible assignments in which exactly two tasks are assigned to trainees.

[3]

Write your answer here...

C

Find the number of possible assignments in which both $X$ and $Y$ are assigned tasks, but neither is assigned either of the first two tasks.

[2]

Write your answer here...

D

Find the number of possible assignments in which at least one trainee is assigned a task, and the first task is not assigned to a trainee.

[3]

Write your answer here...

0

Question 31

HL • Paper 1

Hard

Non Calculator

An examination booklet is to be made from a bank of $15$ questions: $6$ algebra questions, $5$ geometry questions and $4$ probability questions. A booklet contains $7$ questions.

A

I.

Write down the total number of possible booklets if order is not considered.

[1]

Write your answer here...

II.

Find the number of possible booklets containing at least two questions from each of the three areas.

[3]

Write your answer here...

B

booklet satisfying the condition in part (a)(ii) is now arranged in order. Find the number of ordered booklets in which the first question is algebra and the last question is probability.

[4]

Write your answer here...

C

I.

Find the number of ordered booklets counted in part (b) which contain exactly three algebra questions.

[2]

Write your answer here...

II.

Hence find the fraction of the ordered booklets counted in part (b) that contain exactly three algebra questions.

[1]

Write your answer here...

0

Question 32

HL • Paper 1

Hard

Non Calculator

A drama director is assigning $5$ different roles in a play to actors. There are $10$ actors available, consisting of $4$ seniors and $6$ juniors. No actor may receive more than one role.

A

I.

Find the number of possible casts if there are no restrictions.

[2]

Write your answer here...

II.

Find the number of possible casts in which exactly two roles are played by seniors.

[3]

Write your answer here...

B

Two of the five roles are lead roles. Find the number of possible casts in which the two lead roles are played by one senior and one junior, and exactly two of the five roles are played by seniors.

[4]

Write your answer here...

C

One particular senior, Saira, and one particular junior, Jalen, cannot both be in the cast. Find the number of possible casts in which exactly two roles are played by seniors and this restriction is satisfied.

[3]

Write your answer here...

0

Question 33

HL • Paper 1

Hard

Non Calculator

A club has $n$ members, where $n\geq 5$ . For one event, a group of $4$ members is chosen. From this group, a president and a secretary are then chosen; one person cannot hold both roles.

A

I.

Write down an expression, in terms of $n$ , for the number of possible outcomes.

[1]

Write your answer here...

II.

Show that this expression is equal to $n(n-1)\binom{n-2}{2}$ .

[3]

Write your answer here...

B

The number of possible outcomes is $840$ . Find $n$ .

[2]

Write your answer here...

C

For the remainder of the question, take $n=8$ . Two members of the club are Alice and Ben. Find the number of possible outcomes in which Alice is in the group but does not hold a role, and Ben is not in the group.

[2]

Write your answer here...

D

Find the number of possible outcomes in which exactly one of Alice and Ben is in the group, and that person holds a role.

[3]

Write your answer here...

0

Question 34

HL • Paper 1

Hard

Non Calculator

Let $a$ and $b$ be positive integers with $a<b$ . The expansion of

(1+ax)^{-2}(1+bx)

in ascending powers of $x$ begins $1-3x+8x^2+\cdots$ .

A

I.

Find the expansion of $(1+ax)^{-2}$ up to and including the term in $x^2$ .

[2]

Write your answer here...

II.

Find two equations involving $a$ and $b$ .

[2]

Write your answer here...

B

Find the value of $a$ and the value of $b$ .

[3]

Write your answer here...

C

State the restriction on $x$ for which the expansion is valid.

[1]

Write your answer here...

D

Find the coefficient of $x^3$ in the expansion of $(1+4x)^{-2}(1+5x)$ .

[3]

Write your answer here...

0

Question 35

HL • Paper 1

Hard

Non Calculator

Let

g(x)=\frac{\sqrt{1+x}}{1-2x}

A

I.

Write down the expansions of $(1+x)^{\frac12}$ and $(1-2x)^{-1}$ up to and including the term in $x^3$ .

[3]

Write your answer here...

II.

Hence find the expansion of $g(x)$ up to and including the term in $x^3$ .

[2]

Write your answer here...

B

State the restriction on $x$ for which the expansion of $g(x)$ is valid.

[2]

Write your answer here...

C

Use the expansion in part (a)(ii) to approximate $\sqrt{5}$ as a fraction.

[3]

Write your answer here...

0

Question 36

HL • Paper 2

Hard

Calculator Permitted

An exhibition has $10$ different talks available: $4$ science talks, $3$ history talks and $3$ art talks. A programme consists of $6$ of these talks arranged in order in six time slots.

A

I.

Determine the number of possible programmes with no restrictions.

[2]

Write your answer here...

II.

Determine the number of possible programmes which contain exactly one art talk.

[2]

Write your answer here...

B

programme is now required to contain exactly $3$ science talks, $2$ history talks and $1$ art talk. No two science talks may be in consecutive time slots.

I.

Determine the number of possible programmes satisfying these conditions.

[3]

Write your answer here...

II.

Given that a programme satisfies the conditions in part (b), find the probability that the first talk is an art talk.

[3]

Write your answer here...

C

Let $N_k$ be the number of programmes containing exactly $k$ science talks, with no two science talks consecutive. The other talks may be history or art. Determine the value of $k$ for which $N_k$ is greatest.

[3]

Write your answer here...

0

Question 37

HL • Paper 2

Hard

Calculator Permitted

A quality inspector receives a batch of $14$ components. The batch contains $5$ electronic components, $4$ mechanical components and $5$ optical components. A sample of $6$ components is selected without replacement.

A

I.

Find the number of possible samples.

[1]

Write your answer here...

II.

Find the number of samples containing at least one component of each type.

[3]

Write your answer here...

B

The selected components are to be tested one at a time in an ordered sequence.

I.

Find the number of ordered testing sequences using $2$ electronic, $2$ mechanical and $2$ optical components.

[2]

Write your answer here...

II.

Find the number of such ordered sequences in which the two mechanical components are not tested consecutively.

[3]

Write your answer here...

C

sample of $6$ components is chosen at random. Given that it contains at least one component of each type, find the probability that it contains exactly two mechanical components.

[3]

Write your answer here...

0

Question 38

HL • Paper 2

Hard

Calculator Permitted

A gallery has $12$ hooks in a straight line. Eight different posters are to be hung, leaving four hooks empty. Of the posters, $3$ are red and $5$ are blue.

A clean, minimal horizontal row of twelve equally spaced hook icons, labelled 1 through 12 from left to right, with no posters shown. The illustration should make clear that the hooks are in a straight line and that four hooks may remain empty. Do not include coordinate axes, a grid, points, or unrelated labels.

A

I.

Determine the number of ways to hang the $8$ posters if there are no restrictions.

[2]

Write your answer here...

II.

Determine the number of ways to hang the posters if the first and last hooks must be empty.

[2]

Write your answer here...

B

The gallery decides that no two red posters may be on adjacent hooks.

[3]

Write your answer here...

C

Given that no two red posters are on adjacent hooks, find the probability that the first and last hooks are both empty.

[3]

Write your answer here...

0

Question 39

HL • Paper 2

Hard

Calculator Permitted

A music app creates a playlist of $7$ different songs from a library of $12$ songs. The library contains $5$ pop songs, $4$ jazz songs and $3$ classical songs. The order of the playlist matters.

A

I.

Determine the number of possible playlists with no restrictions.

[2]

Write your answer here...

II.

Determine the number of playlists containing exactly $3$ pop songs, $2$ jazz songs and $2$ classical songs.

[2]

Write your answer here...

B

playlist contains exactly $3$ pop songs, $2$ jazz songs and $2$ classical songs. No two classical songs may be consecutive.

I.

Determine the number of playlists counted in part (a)(ii) in which no two classical songs are consecutive.

[3]

Write your answer here...

II.

Find the probability that a playlist counted in part (a)(ii) has no two classical songs consecutive.

[2]

Write your answer here...

C

The app instead chooses an ordered playlist of $7$ different songs uniformly at random from the library. Determine the probability that the playlist contains at least two songs of each genre.

[3]

Write your answer here...

0

Question 40

HL • Paper 2

Hard

Calculator Permitted

Let

f(x)=(1-3x)^{-\frac12}

A

I.

Find the expansion of $f(x)$ up to and including the term in $x^4$ .

[4]

Write your answer here...

II.

State the interval of values of $x$ for which the expansion is valid.

[1]

Write your answer here...

B

Use the expansion from part (a) to estimate

\int_0^{0.1}(1-3x)^{-\frac12}\,\mathrm{d}x

If you did not obtain the expansion in part (a), use $1+\dfrac32x+\dfrac{27}{8}x^2+\dfrac{135}{16}x^3+\dfrac{2835}{128}x^4$ .

[4]

Write your answer here...

C

Use integration to find the exact value of the integral in part (b), and comment on the estimate.

[2]

Write your answer here...

0

Question 41

HL • Paper 3

Hard

Calculator Permitted

A company is forming a project panel from three departments. There are $5$ designers, $4$ analysts and $6$ engineers. A panel consists of $5$ people. The situation is summarized in the table.

Department	Number of people / people
Designers	5
Analysts	4
Engineers	6
Total	15

A

I.

(a)(i) Find the total number of possible panels with no restrictions.

[1]

Write your answer here...

II.

(a)(ii) Find the number of possible panels containing at least one person from each department.

[3]

Write your answer here...

B

I.

(b)(i) For a panel containing at least one person from each department, an engineer is chosen as chair and a designer is chosen as recorder. Find the number of possible outcomes.

[3]

Write your answer here...

C

I.

(c)(i) Suppose instead that the numbers in the three departments are $p$ , $q$ and $r$ . Write down a formula for the number of panels of $5$ people containing at least one person from each department.

[2]

Write your answer here...

II.

(c)(ii) The company later has $n$ people in each department. Find the smallest integer $n$ for which the number of panels of $5$ people containing at least one person from each department is greater than $1000$ .

[3]

Write your answer here...

0

Question 42

HL • Paper 3

Hard

Calculator Permitted

A delivery robot moves on a rectangular grid from $S(0,0)$ to $T(8,5)$ . Each move is either one unit to the right or one unit upwards. Two charging points are $A(3,2)$ and $B(6,3)$ .

Rectangular grid from S to T with charging points A and B.

A

I.

Find the total number of possible paths from $S$ to $T$ .

[1]

Write your answer here...

II.

Find the number of paths from $S$ to $T$ which pass through $A$ .

[2]

Write your answer here...

B

I.

Find the number of paths which pass through both $A$ and $B$ .

[2]

Write your answer here...

II.

Find the number of paths which pass through $A$ but not through $B$ .

[2]

Write your answer here...

C

I.

Find the number of paths which pass through exactly one of $A$ and $B$ .

[3]

Write your answer here...

II.

Hence find the probability that a randomly selected path passes through exactly one of $A$ and $B$ .

[1]

Write your answer here...

III.

For two points $P(a,b)$ and $Q(c,d)$ with $0\leq a\leq c\leq 8$ and $0\leq b\leq d\leq 5$ , write down a formula for the number of paths from $S$ to $T$ which pass through both $P$ and $Q$ .

[2]

Write your answer here...

0

Question 43

HL • Paper 3

Hard

Calculator Permitted

An exhibition has $12$ distinct framed works to be arranged in a straight line. There are $5$ abstract works, $4$ portraits and $3$ landscapes. All works are distinct.

A straight display rail with 12 identical blank positions labelled from left to right; a separate legend identifies the categories abstract, portrait and landscape without assigning categories to particular positions.

A

AI.

Find the total number of possible arrangements.

[1]

Write your answer here...

AII.

Find the number of arrangements in which no two landscapes are adjacent.

[2]

Write your answer here...

B

BI.

Find the probability that no two landscapes are adjacent.

[2]

Write your answer here...

BII.

Find the number of arrangements in which the abstract works are placed only in positions $1,4,7,10$ and $12$ .

[1]

Write your answer here...

C

CI.

Suppose $m$ special works and $n-m$ other works are distinct and arranged in a line. Derive a formula for the number of arrangements in which no two special works are adjacent.

[3]

Write your answer here...

CII.

Using your formula, find the greatest possible value of $m$ when $n=20$ .

[3]

Write your answer here...

0

Question 44

HL • Paper 3

Hard

Calculator Permitted

A secure code has length $8$ . It is formed using $12$ distinct letters, $6$ distinct digits and $4$ distinct symbols. A valid code contains exactly $3$ letters, $3$ digits and $2$ symbols, with no character repeated.

A row of eight blank code positions with three character banks labelled letters, digits and symbols. The digits bank shows exactly six entries: 0, 1, 2, 3, 4 and 5, with each digit appearing once.

A

I.

Find the number of valid codes.

[3]

Write your answer here...

II.

Write the answer to part (a)(i) using permutations rather than combinations.

[1]

Write your answer here...

B

I.

Find the number of valid codes in which at least one symbol occurs among the first four positions.

[3]

Write your answer here...

II.

Find the probability that a randomly selected valid code has at least one symbol among the first four positions.

[1]

Write your answer here...

C

I.

Explain why the two expressions in parts (a)(i) and (a)(ii) are equal.

[2]

Write your answer here...

II.

designer claims that the probability in part (b)(ii) depends on the numbers $12$ , $6$ and $4$ . Justify whether this claim is correct.

[2]

Write your answer here...

0

Question 45

HL • Paper 3

Hard

Calculator Permitted

A simple model for the intensity $I$ of light at a distance $d$ centimetres from a sensor is $I(d)=100\left(1+\frac{d}{25}\right)^{-2}$ for small positive values of $d$ . This question investigates a binomial approximation to the model.

Exact light-intensity curve against distance.

A

I.

Find the binomial expansion of $I(d)$ up to and including the term in $d^3$ .

[3]

Write your answer here...

II.

State the interval of values of $d$ for which this expansion is valid.

[1]

Write your answer here...

B

I.

Use the expansion from part (a) to estimate $I(2)$ .

[2]

Write your answer here...

II.

Calculate the exact value of $I(2)$ and hence find the signed error, defined as estimate minus exact value, using the unrounded estimate from part (b)(i).

[2]

Write your answer here...

C

I.

Use the expansion in part (a) to estimate the average value of $I(d)$ for $0\leq d\leq 3$ .

[3]

Write your answer here...

II.

Using only the terms up to $d^2$ , estimate the smaller positive value of $d$ for which $I(d)=80$ .

[2]

Write your answer here...

0

Question 46

HL • Paper 3

Hard

Calculator Permitted

The extended binomial theorem can be used to approximate cube roots by writing the number in the form $a^3(1+x)$ , where $|x|<1$ . This question compares two approximations for $\sqrt[3]{65}$ .

Rewrite of 65	$x$	$\	x\	$
$65=4^3\left(1+\frac{1}{64}\right)$	$\frac{1}{64}$	$\frac{1}{64}$
$65=5^3\left(1-\frac{60}{125}\right)$	$-\frac{60}{125}$	$\frac{60}{125}$

A

I.

Find the first four non-zero terms in the expansion of $(1+x)^{\frac13}$ .

[3]

Write your answer here...

B

I.

Use $65=4^3\left(1+\frac{1}{64}\right)$ and the expansion in part (a) to approximate $\sqrt[3]{65}$ . Give your answer to three significant figures, retaining the unrounded value for use in part (d).

[3]

Write your answer here...

C

I.

Use $65=5^3\left(1-\frac{60}{125}\right)$ and the same number of terms to approximate $\sqrt[3]{65}$ .

[2]

Write your answer here...

II.

Explain which of the two approximations should be more accurate, without using the exact cube root.

[1]

Write your answer here...

D

I.

Use a GDC to obtain a decimal approximation to $\sqrt[3]{65}$ to sufficient accuracy, and find the absolute error in the unrounded four-term approximation from part (b)(i).

[3]

Write your answer here...

0

Question 47

HL • Paper 1

Hard

Non Calculator

Consider the expansion of

(1-3x)^{-3}

A

I.

Find the expansion up to and including the term in $x^4$ .

[4]

Write your answer here...

II.

State the restriction on $x$ for which the expansion is valid.

[1]

Write your answer here...

B

Find the coefficient of $x^4$ in the expansion of $(1+x)^2(1-3x)^{-3}$ .

[3]

Write your answer here...

C

Show that the coefficient of $x^n$ in $(1-3x)^{-3}$ is $\dfrac{(n+1)(n+2)}{2}3^n$ , for $n\geq 0$ .

[2]

Write your answer here...

D

Hence find, in terms of $n$ , the coefficient of $x^n$ in $(1+x)^2(1-3x)^{-3}$ , where $n\geq 2$ .

[3]

Write your answer here...

0

Question 48

HL • Paper 1

Hard

Non Calculator

The binomial expansion of $(1+cx)^r$ , where $c$ and $r$ are non-zero rational constants, begins

1-6x+24x^2+\cdots

A

I.

Write down two equations involving $c$ and $r$ .

[2]

Write your answer here...

II.

Find the value of $c$ and the value of $r$ .

[3]

Write your answer here...

B

Find the coefficient of $x^3$ in the expansion.

[2]

Write your answer here...

C

State the restriction on $x$ for which the expansion is valid.

[1]

Write your answer here...

D

Use the expansion up to and including the term in $x^3$ to approximate $\left(\dfrac65\right)^{-3}$ as a fraction.

[3]

Write your answer here...

0

Question 49

HL • Paper 2

Hard

Calculator Permitted

A club has $7$ junior members and $8$ senior members. A project group of $5$ members is to be selected.

A

I.

Write down an expression, as a sum, for the number of possible project groups classified by the number of junior members.

[2]

Write your answer here...

II.

Hence show that $\displaystyle \sum_{k=0}^{5}\binom7k\binom8{5-k}=\binom{15}{5}$ .

[2]

Write your answer here...

B

After a project group is selected, a coordinator and a recorder are chosen from the group. The two roles must be held by members from different age groups. An outcome consists of the selected project group together with the assigned coordinator and recorder.

I.

Find the number of possible outcomes if the project group contains exactly $k$ junior members.

[2]

Write your answer here...

II.

Find the total number of possible outcomes.

[3]

Write your answer here...

C

project group is selected at random. Given that it contains at least one junior member and at least one senior member, find the probability that it contains more juniors than seniors.

[3]

Write your answer here...

0

Question 50

HL • Paper 2

Hard

Calculator Permitted

Consider the function

f(x)=\left(1-\frac{x}{3}\right)^{-2}(1+2x)^{\frac12}

A

I.

Write down the binomial expansion of $\left(1-\dfrac{x}{3}\right)^{-2}$ up to and including the term in $x^3$ .

[2]

Write your answer here...

II.

Write down the binomial expansion of $(1+2x)^{\frac12}$ up to and including the term in $x^3$ .

[2]

Write your answer here...

B

Hence find the expansion of $f(x)$ up to and including the term in $x^3$ .

[3]

Write your answer here...

C

C.

State the interval of values of $x$ for which both binomial expansions used in part (a) are valid.

[2]

Write your answer here...

D

Use the cubic expansion from part (b) to estimate the positive solution of $f(x)=1.2$ in the interval identified in part (c). If you did not obtain the expansion in part (b), use $1+\dfrac53x+\dfrac12x^2+\dfrac{35}{54}x^3$ .

[4]

Write your answer here...

0

Question 51

HL • Paper 2

Hard

Calculator Permitted

The expansion of

(1+px)^{-\frac12}(1+qx)^{-1}

in ascending powers of $x$ begins

1-5x+21x^2+\cdots

where $p$ and $q$ are positive integers.

A

I.

Show that $p$ and $q$ satisfy $\dfrac p2+q=5$ .

[2]

Write your answer here...

II.

Determine the value of $p$ and the value of $q$ .

[4]

Write your answer here...

B

Using the values of $p$ and $q$ , find the coefficient of $x^3$ in the expansion.

[3]

Write your answer here...

C

State the interval of values of $x$ for which the expansion is valid.

[3]

Write your answer here...

0

Question 52

HL • Paper 2

Hard

Calculator Permitted

Let $a_r$ be the coefficient of $x^r$ in the expansion of

(1-2x)^{-3}

A

I.

Find $a_0$ , $a_1$ , $a_2$ and $a_3$ .

[2]

Write your answer here...

II.

Show that $a_r=2^r\binom{r+2}{2}$ for $r\geq0$ .

[3]

Write your answer here...

B

Use a GDC to determine the smallest value of $r$ for which $a_r>50000$ .

[3]

Write your answer here...

C

The first $n+1$ terms of the expansion are used to approximate $(1-2x)^{-3}$ at $x=0.1$ . Determine the smallest value of $n$ for which the absolute error is less than $0.001$ .

[4]

Write your answer here...

0

Question 53

HL • Paper 2

Hard

Calculator Permitted

A school fair has $9$ different game stalls and $6$ different food stalls. A visitor plans to visit $7$ stalls in a particular order, without visiting any stall more than once. Assume that each eligible ordered plan is equally likely whenever a probability is requested.

A

I.

Find the number of possible ordered plans with no restrictions.

[2]

Write your answer here...

II.

Find the number of ordered plans containing exactly $4$ game stalls and $3$ food stalls.

[2]

Write your answer here...

B

visitor chooses exactly $4$ game stalls and $3$ food stalls. The visitor does not want to visit two food stalls consecutively.

I.

Find the number of possible ordered plans with exactly $4$ game stalls and $3$ food stalls, with no two food stalls consecutive.

[3]

Write your answer here...

II.

Given the visitor chooses exactly $4$ game stalls and $3$ food stalls, find the probability that no two food stalls are consecutive.

[2]

Write your answer here...

C

visitor chooses exactly $4$ game stalls and $3$ food stalls, with no two food stalls consecutive. Find the probability that the first and last stalls visited are both game stalls.

[5]

Write your answer here...

0

Question 54

HL • Paper 3

Hard

Calculator Permitted

In a lottery, a ticket consists of $7$ different numbers chosen from the integers $1$ to $50$ . The order of the numbers does not matter. A draw selects $7$ winning numbers. The prize table depends only on the number of winning numbers matched by a ticket. Tickets matching fewer than $5$ winning numbers receive no prize.

Exact matches	Prize [dollars]
5	20
6	1000
7	1000000

A

I.

Find the number of different tickets possible.

[1]

Write your answer here...

II.

Find the probability that one ticket matches all $7$ winning numbers.

[2]

Write your answer here...

B

I.

Find the number of tickets which match exactly $5$ of the winning numbers.

[2]

Write your answer here...

II.

Find the number of tickets which match at least $5$ of the winning numbers.

[2]

Write your answer here...

C

I.

The prizes for exactly $5$ , exactly $6$ and exactly $7$ matches are respectively $20, $1000 and $1000000. Find the expected prize value of one ticket.

[3]

Write your answer here...

II.

ticket costs $2. Comment on the fairness of the lottery using your answer to part (c)(i).

[1]

Write your answer here...

D

I.

For this lottery, write down a formula for the number of tickets which match exactly $k$ of the winning numbers, where $0\leq k\leq 7$ .

[2]

Write your answer here...

II.

Explain why the formula in part (d)(i) is consistent with the total number of tickets.

[1]

Write your answer here...

0

Question 55

HL • Paper 3

Hard

Calculator Permitted

A foundation awards research grants to applicants from three fields. There are $4$ applicants in algebra, $3$ in geometry and $3$ in statistics. First a shortlist of $4$ applicants is chosen. Then first, second and third grant places are awarded to three different shortlisted applicants.

Field / stage	Count / rule
Total applicants	10
Algebra	4
Geometry	3
Statistics	3
Shortlist	Choose 4 applicants
Grant places	Award 1st, 2nd and 3rd places to 3 different shortlisted applicants

A

I.

Find the number of possible outcomes with no field restriction.

[2]

Write your answer here...

B

I.

Find the number of shortlists which contain at least one applicant from each field.

[3]

Write your answer here...

II.

Hence find the number of possible outcomes if the shortlist must contain at least one applicant from each field.

[1]

Write your answer here...

C

I.

Find the number of possible outcomes in which the three grant winners include exactly one applicant from each field.

[3]

Write your answer here...

II.

Find the probability that the three grant winners include exactly one applicant from each field, given that there is no field restriction.

[1]

Write your answer here...

D

I.

If there are $x$ applicants in each of the three fields, find the smallest integer $x$ for which the number of shortlists of $4$ containing at least one applicant from each field exceeds $1000$ .

[3]

Write your answer here...

0

Question 56

HL • Paper 3

Hard

Calculator Permitted

Consider the function $f(x)=(1-2x)^{-\frac12}(1+x)^{-2}$ This question uses binomial expansions to approximate an area under the curve.

Curve of f(x) on [0,0.2] with the interval endpoints marked.

A

I.

Expand $(1-2x)^{-\frac12}$ up to and including the term in $x^3$ .

[2]

Write your answer here...

II.

Expand $(1+x)^{-2}$ up to and including the term in $x^3$ .

[2]

Write your answer here...

III.

State the interval of values of $x$ for which both expansions are valid.

[1]

Write your answer here...

B

I.

Hence find the expansion of $f(x)$ up to and including the term in $x^3$ .

[3]

Write your answer here...

C

I.

Use the expansion from part (b) to estimate $\int_0^{0.2} f(x)\,\mathrm{d}x$ .

[3]

Write your answer here...

D

I.

Write down a summation expression for the coefficient of $x^n$ in the expansion of $f(x)$ , where $n\geq 0$ .

[3]

Write your answer here...

0

Question 57

HL • Paper 3

Hard

Calculator Permitted

A correction factor is modelled by $F(x)=(1-ax)^{-\frac13}+bx$ where $a$ and $b$ are constants and $a>0$ . The expansion of $F(x)$ begins $F(x)=1+5x+8x^2+\cdots$

Power	Symbolic coefficient in $F(x)$	Target coefficient
$x^0$	1	1
$x^1$	$\frac{a}{3}+b$	5
$x^2$	$\frac{2a^2}{9}$	8

A

I.

Expand $(1-ax)^{-\frac13}$ up to and including the term in $x^3$ .

[3]

Write your answer here...

II.

Write down the restriction on $x$ in terms of $a$ .

[1]

Write your answer here...

B

I.

Determine the values of $a$ and $b$ .

[3]

Write your answer here...

II.

For these values of $a$ and $b$ , write down the interval of validity of the expansion.

[1]

Write your answer here...

C

I.

Find the coefficient of $x^3$ in the expansion of $F(x)$ .

[2]

Write your answer here...

II.

Use terms up to and including $x^3$ to approximate $F(0.05)$ .

[2]

Write your answer here...

III.

Explain why the approximation in part (c)(ii) is valid.

[1]

Write your answer here...

0

Question 58

HL • Paper 3

Hard

Calculator Permitted

A cafe designs snack boxes. There are $4$ fruits, $5$ nuts and $7$ toppings available. A snack box contains exactly $6$ different items. Generating functions can be used to represent the counting.

Category	Number available	Generating-function factor
Fruits	4	$(1+x)^4$
Nuts	5	$(1+x)^5$
Toppings	7	$(1+x)^7$
All categories	16	$(1+x)^4(1+x)^5(1+x)^7$

A

I.

Explain why the coefficient of $x^6$ in $(1+x)^4(1+x)^5(1+x)^7$ gives the total number of snack boxes.

[2]

Write your answer here...

II.

Find the total number of snack boxes.

[1]

Write your answer here...

III.

Find the number of snack boxes containing at least one item from each category.

[2]

Write your answer here...

B

I.

Find the number of snack boxes containing exactly two fruits.

[2]

Write your answer here...

II.

Interpret the calculation in part (b)(i) in terms of the generating function.

[1]

Write your answer here...

C

I.

Find the number of snack boxes containing an even number of nuts, with no restriction on whether the other categories are represented.

[3]

Write your answer here...

II.

Give a generating-function expression whose coefficient of $x^6$ also gives the answer to part (c)(i).

[1]

Write your answer here...

D

I.

customer says that choosing $6$ items from $16$ is the same model as arranging $6$ items in a row. Explain the error in this statement.

[2]

Write your answer here...

0

Question 59

HL • Paper 3

Hard

Calculator Permitted

An engineer uses a linear correction factor to improve the binomial approximation of $G_m(x)=(1+kx)(1-x)^{-m}$ where $m\in\mathbb{Q}$ , $m>0$ , and $|x|<1$ .

Term	In $(1-x)^{-m}$	In $G_m(x)$
constant	to be found	to be found
$x$	to be found	to be found
$x^2$	to be found	to be found

A

I.

Expand $(1-x)^{-m}$ up to and including the term in $x^2$ .

[2]

Write your answer here...

II.

Hence write the expansion of $G_m(x)$ up to and including the term in $x^2$ .

[2]

Write your answer here...

B

I.

Find $k$ in terms of $m$ so that the coefficient of $x$ in $G_m(x)$ is zero.

[2]

Write your answer here...

II.

For this value of $k$ , find the coefficient of $x^2$ in $G_m(x)$ and simplify it.

[2]

Write your answer here...

C

I.

Take $m=\frac12$ . Write down $G_{\frac12}(x)$ using the value of $k$ found in part (b), and find its expansion up to and including the term in $x^2$ .

[3]

Write your answer here...

D

I.

Use the result from part (c) to approximate $G_{\frac12}(0.1)$ , and explain why the approximation is expected to be close to $1$ .

[2]

Write your answer here...

0

Question 60

HL • Paper 2

Hard

Calculator Permitted

Consider

g(x)=(2+x)^{-\frac32}(1-4x)^{-1}

A

I.

Rewrite $(2+x)^{-\frac32}$ in the form $A(1+Bx)^{-\frac32}$ , stating the values of $A$ and $B$ .

[2]

Write your answer here...

II.

Find the expansion of $(2+x)^{-\frac32}$ up to and including the term in $x^2$ .

[3]

Write your answer here...

B

Find the expansion of $g(x)$ up to and including the term in $x^2$ .

[3]

Write your answer here...

C

State the interval of values of $x$ for which the expansion in part (b) is valid.

[2]

Write your answer here...

D

Use the quadratic expansion from part (b) to estimate the value of $x$ in the interval of validity for which $g(x)=0.5$ . If you did not obtain the expansion in part (b), use $\dfrac{1}{2\sqrt2}\left(1+\dfrac{13x}{4}+\dfrac{431x^2}{32}\right)$ .

[3]

Write your answer here...

0