Consider the binomial expansion of $\left(\frac{ax}{2} + \frac{3}{x^2}\right)^5$ . Find an expression, in terms of $a$ , for the coefficient of the $x^{-1}$ term. [5 marks] Given that the coefficient of the $x^{-1}$ term is exactly $90$ , find the value of the constant $a$ . [2 marks]

(a) General term: $\binom{5}{r}\left(\frac{ax}{2}\right)^{5-r}\left(\frac{3}{x^2}\right)^r$ (M1) Power of $x$ : $(5 - r) - 2r = -1 \Rightarrow r = 2$ (A1) Substitute $r = 2$ : $\binom{5}{2}\left(\frac{a}{2}\right)^3(3)^2 = 10 \times \frac{a^3}{8} \times 9$ (M1) $= \frac{90a^3}{8}$ (A1) $= \frac{45a^3}{4}$ A1 (b) Set $\frac{45a^3}{4} = 90 \Rightarrow a^3 = 8$ (M1) $a = 2$ A1

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IB Maths AA HL · Unit 1: Number and Algebra

IB Maths AA HL Counting and Binomials Questions

Q: How many IB Maths AA HL counting and binomials questions are there?

There are 22 exam-style counting and binomials questions in the AA HL question bank (Paper 1: 19 · Paper 2: 3), graded 5 easy, 8 medium, 9 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is counting and binomials on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 19 · Paper 2: 3. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Q: Where can I get the mark schemes?

Open the AA HL Unit 1 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

Updated October 2026

Exam-style IB Maths AA HL counting and binomials questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

22 questions
Paper 1: 19
Paper 2: 3
5 easy
8 medium
9 hard
3 worked examples

Practise Counting and Binomials questions →

What you need to know

HL AA extends SL AA's binomial to (1+x)ⁿ for any real n, needing |x| < 1 for convergence. Paper 3 combines this with Maclaurin series. Binomial theorem for rational and negative exponents overview →

What's examined in AA HL counting and binomials

The question bank covers these counting and binomials question types (number of questions in brackets):

General Binomial Theorem (11)
Counting Principles (8)
Integer Power Binomials (3)

Counting and Binomials worked examples

Worked example 1: Evaluating combinations with constraints · easy

A committee of $4$ people is to be chosen from a group consisting of $6$ men and $5$ women. Find the number of different committees that can be formed if the committee must contain exactly $2$ men and $2$ women.

Solution

5 steps

Examiner tip: Students frequently confuse permutations with combinations; look out for keywords like "committee" or "group" which universally indicate that the internal order of selection is irrelevant and combinations should be used.

Worked example 2: Finding a specific binomial expansion term · medium

Find the exact coefficient of the term in $x^{7}$ in the binomial expansion of ${(2 x - \frac{1}{x^{2}})}^{10}$ .

Solution

6 steps · 1 to fill in

Examiner tip: A classic trap is neglecting to carry the negative sign into the general term formula, which reverses the sign of the final answer whenever

r

resolves to an odd integer.

Worked example 3: Expanding with negative fractional indices · hard

Find the first three terms, in ascending powers of $x$ , in the Maclaurin expansion of $(1 + 3 x)^{- 2}$ , and state the range of values of $x$ for which this expansion is valid.

Solution

6 steps

Examiner tip: Students easily forget that when substituting into the extended binomial theorem, they must square the entire secondary term (including the coefficient, meaning

(3 x)^{2}

becomes

9 x^{2}

, not

3 x^{2}

Try these IB Maths AA HL counting and binomials questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 2 marks · Paper 1

Evaluate $(\binom{7}{3})$ without a calculator and state exactly what this value represents in the context of combinatorics.

Attempt it and see the mark scheme →

Question 2 · medium · 7 marks · Paper 1

Consider the binomial expansion of ${(\frac{a x}{2} + \frac{3}{x^{2}})}^{5}$ .

Find an expression, in terms of $a$ , for the coefficient of the $x^{- 1}$ term. [5 marks]
Given that the coefficient of the $x^{- 1}$ term is exactly $90$ , find the value of the constant $a$ . [2 marks]

Full worked solution Free, no sign-in

(a) General term: $(\binom{5}{r}) {(\frac{a x}{2})}^{5 - r} {(\frac{3}{x^{2}})}^{r}$ (M1)
Power of $x$ : $(5 - r) - 2 r = - 1 \Rightarrow r = 2$ (A1)
Substitute $r = 2$ : $(\binom{5}{2}) {(\frac{a}{2})}^{3} (3)^{2} = 10 \times \frac{a^{3}}{8} \times 9$ (M1)
$= \frac{90 a^{3}}{8}$ (A1)
$= \frac{45 a^{3}}{4}$ A1
(b) Set $\frac{45 a^{3}}{4} = 90 \Rightarrow a^{3} = 8$ (M1)
$a = 2$ A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 2

Consider the expansion of $(1 - 3 x)^{4} (1 - 2 k x)^{2}$ , where $k$ is a constant.

Given that the coefficient of the $x^{6}$ term is $36$ , find the possible values of $k$ .

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AA HL counting and binomials questions are there?

There are 22 exam-style counting and binomials questions in the AA HL question bank (Paper 1: 19 · Paper 2: 3), graded 5 easy, 8 medium, 9 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is counting and binomials on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 19 · Paper 2: 3. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Where can I get the mark schemes?

Open the AA HL Unit 1 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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Teaching this topic?

Free ready-to-teach lessons (slides and teacher notes): Binomial theorem for rational / negative n · Counting principles: permutations & combinations.

IB Maths AA HL Counting and Binomials Questions

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What you need to know

What's examined in AA HL counting and binomials

Counting and Binomials worked examples

Worked example 1: Evaluating combinations with constraints · easy

Solution

Worked example 2: Finding a specific binomial expansion term · medium

Solution

Worked example 3: Expanding with negative fractional indices · hard

Solution

Try these IB Maths AA HL counting and binomials questions

Question 1 · easy · 2 marks · Paper 1

Question 2 · medium · 7 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 2

Revise counting and binomials for the exam

FAQ

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