IB Maths AA HL · Unit 1: Number and Algebra
IB Maths AA HL Counting and Binomials Questions
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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 19 questions
- Paper 1: 15
- Paper 2: 4
- 5 easy
- 6 medium
- 8 hard
- 3 worked examples
Practise Counting and Binomials questions →
AA HL formula booklet
What's examined in AA HL counting and binomials
The question bank covers these counting and binomials question types (number of questions in brackets):
- Counting Principles (8)
- Integer Power Binomials (6)
- General Binomial Theorem (5)
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.
1. Identify that the selection order within a committee does not matter, meaning combinations (nCr) should be used.
2. Calculate the number of ways to independently choose $2$ men from $6$: $\binom{6}{2} = \frac{6!}{2!(6-2)!} = 15$.
3. Calculate the number of ways to independently choose $2$ women from $5$: $\binom{5}{2} = \frac{5!}{2!(5-2)!} = 10$.
4. Apply the fundamental counting principle by multiplying the number of options for these independent events together.
5. Evaluate the final product: $15 \times 10 = \mathbf{150}$.
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 $\left(2x - \frac{1}{x^2}\right)^{10}$.
1. Write the general term formula for the binomial expansion: $T_{r+1} = \binom{10}{r} (2x)^{10-r} \left(-x^{-2}\right)^r$.
2. Group the constants and the powers of $x$ separately: $\binom{10}{r} 2^{10-r} (-1)^r x^{10-r} x^{-2r}$.
3. Combine the indices of $x$ using exponent laws: $x^{10 - r - 2r} = x^{10 - 3r}$.
4. Equate this combined exponent to the desired power to find $r$: $10 - 3r = 7 \implies 3r = 3 \implies r = 1$.
5. Substitute $r = 1$ back into the coefficient part of the general term: $\binom{10}{1} 2^{10-1} (-1)^1$.
6. Evaluate the final exact coefficient: $10 \times 512 \times -1 = \mathbf{-5120}$.
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 $(1 + 3x)^{-2}$, and state the range of values of $x$ for which this expansion is valid.
1. Recognize the need for the extended binomial theorem from the formula booklet since the exponent $n = -2$ is a negative integer.
2. Identify the standard formula terms where $(1 + y)^n \approx 1 + ny + \frac{n(n-1)}{2!}y^2$, applying $y = 3x$.
3. Substitute the values into the formula: $1 + (-2)(3x) + \frac{(-2)(-3)}{2}(3x)^2$.
4. Expand the squares and products carefully: $1 - 6x + \frac{6}{2}(9x^2)$.
5. Simplify to state the first three terms: $\mathbf{1 - 6x + 27x^2}$.
6. Determine the domain of validity, which requires $|y| < 1$, hence $|3x| < 1$, giving the range $\mathbf{-\frac{1}{3} < x < \frac{1}{3}}$.
Examiner tip: Students easily forget that when substituting into the extended binomial theorem, they must square the entire secondary term (including the coefficient, meaning $(3x)^2$ becomes $9x^2$, not $3x^2$).
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.
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.
Question 2 · medium · 7 marks · Paper 1
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]
Question 3 · hard · 6 marks · Paper 2
Consider the expansion of \((1 - 3x)^4(1 - 2kx)^2\), where \(k\) is a constant.
Given that the coefficient of the \(x^6\) term is \(36\), find the possible values of \(k\).
All 19 counting and binomials questions with mark schemes →
FAQ
How many IB Maths AA HL counting and binomials questions are there?
There are 19 exam-style counting and binomials questions in the AA HL question bank (Paper 1: 15 · Paper 2: 4), graded 5 easy, 6 medium, 8 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: 15 · Paper 2: 4. 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.
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