Counting Principles - IB Math AA Question Bank

Question 1

HL • Paper 1

Easy

Non Calculator

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]

B

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

[3]

Question 2

HL • Paper 1

Easy

Non Calculator

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]

B

Find the value of $n$ .

[2]

Question 3

HL • Paper 2

Easy

Calculator Permitted

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]

B

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

[2]

C

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

[3]

Question 4

HL • Paper 2

Easy

Calculator Permitted

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]

B

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

[3]

Question 5

HL • Paper 2

Easy

Calculator Permitted

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]

B

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

[1]

Question 6

HL • Paper 1

Medium

Non Calculator

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]

B

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

[3]

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]

Question 7

HL • Paper 1

Medium

Non Calculator

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]

B

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

[2]

C

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

[2]

Question 8

HL • Paper 1

Medium

Non Calculator

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]

B

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

[2]

Question 9

HL • Paper 1

Medium

Non Calculator

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]

B

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

[3]

C

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

[2]

Question 10

HL • Paper 1

Medium

Non Calculator

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]

B

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

[2]

C

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

[2]

Question 11

HL • Paper 1

Medium

Non Calculator

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]

B

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

[1]

C

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

[1]

Question 12

HL • Paper 2

Medium

Calculator Permitted

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]

B

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

[4]

Question 13

HL • Paper 2

Medium

Calculator Permitted

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]

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]

Question 14

HL • Paper 2

Medium

Calculator Permitted

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]

B

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

[2]

C

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

[2]

Question 15

HL • Paper 2

Medium

Calculator Permitted

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]

B

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

[2]

Question 16

HL • Paper 2

Medium

Calculator Permitted

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]

B

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

[2]

Question 17

HL • Paper 1

Medium

Non Calculator

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]

B

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

[2]

C

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

[4]

Question 18

HL • Paper 1

Medium

Non Calculator

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]

B

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

[4]

Question 19

HL • Paper 1

Medium

Non Calculator

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]

B

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

[2]

Question 20

HL • Paper 1

Medium

Non Calculator

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]

B

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

[1]

C

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

[2]

Question 21

HL • Paper 2

Medium

Calculator Permitted

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]

B

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

[3]

Question 22

HL • Paper 2

Medium

Calculator Permitted

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]

B

Determine the value of $c$ .

[2]

C

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

[1]

Question 23

HL • Paper 2

Medium

Calculator Permitted

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]

B

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

[3]

Question 24

HL • Paper 1

Medium

Non Calculator

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]

II.

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

[2]

B

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

[3]

C

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

[4]

Question 25

HL • Paper 1

Medium

Non Calculator

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]

II.

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

[3]

B

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

[3]

C

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

[2]

Question 26

HL • Paper 1

Medium

Non Calculator

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]

II.

Find $q$ and $r$ .

[3]

B

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

[1]

C

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

[3]

Question 27

HL • Paper 2

Medium

Calculator Permitted

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]

II.

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

[1]

B

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

[2]

C

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

[3]

D

Explain why the binomial expansion is appropriate for this approximation.

[1]

Question 28

HL • Paper 2

Medium

Calculator Permitted

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]

B

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

[2]

Question 29

HL • Paper 1

Hard

Non Calculator

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]

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]

B

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

[4]

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]

Question 30

HL • Paper 1

Hard

Non Calculator

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]

II.

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

[2]

B

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

[3]

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]

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]

Question 31

HL • Paper 1

Hard

Non Calculator

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]

II.

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

[3]

B

A 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]

C

I.

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

[2]

II.

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

[1]

Question 32

HL • Paper 1

Hard

Non Calculator

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]

II.

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

[3]

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]

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]

Question 33

HL • Paper 1

Hard

Non Calculator

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]

II.

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

[3]

B

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

[2]

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]

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]

Question 34

HL • Paper 1

Hard

Non Calculator

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]

II.

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

[2]

B

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

[3]

C

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

[1]

D

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

[3]

Question 35

HL • Paper 1

Hard

Non Calculator

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]

II.

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

[2]

B

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

[2]

C

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

[3]

Question 36

HL • Paper 2

Hard

Calculator Permitted

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]

II.

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

[2]

B

A 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]

II.

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

[3]

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]

Question 37

HL • Paper 2

Hard

Calculator Permitted

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]

II.

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

[3]

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]

II.

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

[3]

C

A 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]

Question 38

HL • Paper 2

Hard

Calculator Permitted

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]

II.

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

[2]

B

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

[3]

C

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

[3]

Question 39

HL • Paper 2

Hard

Calculator Permitted

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]

II.

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

[2]

B

A 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]

II.

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

[2]

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]

Question 40

HL • Paper 2

Hard

Calculator Permitted

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]

II.

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

[1]

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]

C

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

[2]

Question 41

HL • Paper 3

Hard

Calculator Permitted

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]

II.

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

[3]

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]

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]

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]

Question 42

HL • Paper 3

Hard

Calculator Permitted

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]

II.

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

[2]

B

I.

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

[2]

II.

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

[2]

C

I.

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

[3]

II.

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

[1]

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]

Question 43

HL • Paper 3

Hard

Calculator Permitted

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]

AII.

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

[2]

B

BI.

Find the probability that no two landscapes are adjacent.

[2]

BII.

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

[1]

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]

CII.

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

[3]

Question 44

HL • Paper 3

Hard

Calculator Permitted

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]

II.

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

[1]

B

I.

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

[3]

II.

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

[1]

C

I.

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

[2]

II.

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

[2]

Question 45

HL • Paper 3

Hard

Calculator Permitted

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]

II.

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

[1]

B

I.

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

[2]

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]

C

I.

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

[3]

II.

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

[2]

Question 46

HL • Paper 3

Hard

Calculator Permitted

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]

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]

C

I.

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

[2]

II.

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

[1]

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]

Question 47

HL • Paper 1

Hard

Non Calculator

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]

II.

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

[1]

B

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

[3]

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]

D

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

[3]

Question 48

HL • Paper 1

Hard

Non Calculator

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]

II.

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

[3]

B

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

[2]

C

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

[1]

D

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

[3]

Question 49

HL • Paper 2

Hard

Calculator Permitted

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]

II.

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

[2]

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]

II.

Find the total number of possible outcomes.

[3]

C

A 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]

Question 50

HL • Paper 2

Hard

Calculator Permitted

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]

II.

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

[2]

B

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

[3]

C

C.

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

[2]

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]

Question 51

HL • Paper 2

Hard

Calculator Permitted

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]

II.

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

[4]

B

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

[3]

C

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

[3]

Question 52

HL • Paper 2

Hard

Calculator Permitted

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]

II.

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

[3]

B

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

[3]

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]

Question 53

HL • Paper 2

Hard

Calculator Permitted

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]

II.

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

[2]

B

A 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]

II.

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

[2]

C

A 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]

Question 54

HL • Paper 3

Hard

Calculator Permitted

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]

II.

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

[2]

B

I.

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

[2]

II.

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

[2]

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]

II.

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

[1]

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]

II.

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

[1]

Question 55

HL • Paper 3

Hard

Calculator Permitted

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]

B

I.

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

[3]

II.

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

[1]

C

I.

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

[3]

II.

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

[1]

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]

Question 56

HL • Paper 3

Hard

Calculator Permitted

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]

II.

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

[2]

III.

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

[1]

B

I.

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

[3]

C

I.

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

[3]

D

I.

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

[3]

Question 57

HL • Paper 3

Hard

Calculator Permitted

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]

II.

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

[1]

B

I.

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

[3]

II.

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

[1]

C

I.

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

[2]

II.

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

[2]

III.

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

[1]

Question 58

HL • Paper 3

Hard

Calculator Permitted

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]

II.

Find the total number of snack boxes.

[1]

III.

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

[2]

B

I.

Find the number of snack boxes containing exactly two fruits.

[2]

II.

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

[1]

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]

II.

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

[1]

D

I.

A 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]

Question 59

HL • Paper 3

Hard

Calculator Permitted

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]

II.

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

[2]

B

I.

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

[2]

II.

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

[2]

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]

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]

Question 60

HL • Paper 2

Hard

Calculator Permitted

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]

II.

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

[3]

B

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

[3]

C

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

[2]

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]