AQA AS Level Mathematics Paper 1, June 2025: Question 12

Mark scheme

Show the mark scheme

Q Marking instructions AO Marks Typical solution

12(a) Writes 22 1.1b B1 22

Subtotal 1

12(b) Sets up correct factorial 3.1a M1 40! 40! 40! 17 23!!

expressions as part of a division ÷ = ×

18 22!! 17 23!! 18 22!! 40!

or multiplication

17 23!!

Completes a reasoned 1.1b R1 =

23 18 22!!

argument to obtain 23

18 =

Subtotal 2

How to answer it

Binomial Coefficients & Factorial Notation

📌 What this question tests

This question assesses your foundational understanding of the formal factorial definition of the combination formula ⁿCᵣ and your algebraic skill in simplifying ratios of factorials without relying on a calculator.

The definition: ⁿCᵣ = n! / (r!(n - r)!)
Division of algebraic fractions involving factorials
Factorial cancellation rules: n! = n × (n - 1)!

Question 12 (a)

Finding the unknown value in a factorial combination formula [1 Mark]

💡 Key Knowledge

The standard formula for combinations is:

ⁿCᵣ = n! / (r!(n - r)!)

Comparing ⁴⁰C₁₈ = 40! / (18! p!) directly with the formula:

n = 40
r = 18
p = n - r = 40 - 18

📐 Calculation

Identify the formula component: p = 40 - 18
Subtract to find p: p = 22

✅ Correct Answer

p = 22

B1 (1.1b): Correct value of 22 stated clearly. Writing 22! is incorrect—the exclamation mark is already given in the question.

❌ Common Errors

Writing 22! instead of 22. The question states p! , so p = 22 .
Arithmetic slip when calculating 40 - 18 (e.g. obtaining 32 or 28).

Question 12 (b)

Simplifying a ratio of consecutive binomial coefficients [2 Marks]

💡 Key Knowledge

To divide fractions, multiply by the reciprocal:

(A / B) ÷ (C / D) = (A / B) × (D / C)

Unpack factorials using their recursive property:

23! = 23 × 22!, so 23! / 22! = 23
18! = 18 × 17!, so 17! / 18! = 1 / 18

🧠 Exam Technique: "Show that" Proofs

Because the numerator 23 is already given in the question, you must show full, convincing algebraic working:

Write out both expressions fully using factorials.
Invert the second fraction and multiply.
Explicitly show the cancellation of 40! and how 23!/22! reduces to 23 and 17!/18! reduces to 18.
State the final integer: q = 18.

📐 Step-by-Step Calculation

Set up the expressions using the formula:
⁴⁰C₁₈ = 40! / (18! 22!)
⁴⁰C₁₇ = 40! / (17! (40 - 17)!) = 40! / (17! 23!)
Set up the division and change to multiplication:
⁴⁰C₁₈ / ⁴⁰C₁₇ = (40! / (18! 22!)) ÷ (40! / (17! 23!))
= (40! / (18! 22!)) × ((17! 23!) / 40!)
Cancel common terms:
Cancel 40!: = (17! 23!) / (18! 22!)
Expand the larger factorials: = (17! × 23 × 22!) / (18 × 17! × 22!)
Simplify the fraction:
= 23 / 18
Identify the integer q:
Comparing 23 / 18 to 23 / q gives q = 18.

✅ Final Solution & Marks

⁴⁰C₁₈ / ⁴⁰C₁₇ = 23 / 18 ⇒ q = 18

M1 (3.1a): Sets up correct factorial expressions as part of a division or multiplication, showing the reciprocal.
R1 (1.1b): Completes a reasoned argument to obtain 23/18 (or explicitly writes q = 18).

❌ Common Errors

Using a calculator directly: Evaluating ⁴⁰C₁₈ and ⁴⁰C₁₇ as huge decimal numbers or integers scores 0 marks. The command word is "Use the result from part (a) to show that...".
Jumping steps: Writing the initial fraction and immediately jumping to 23/18 without showing the cancellation of factorials loses the reasoning mark (R1).
Forgetting to identify q: Always conclude by making sure your answer explicitly matches the requested form.

Topics

Pure Mathematics · D: Sequences and series