A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 8: The binomial expansion · Lesson 1

Pascal's triangle and factorial notation

70 marks

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Date

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Quick questions to start.

A1
Last lesson
Prove that the sum of any two consecutive odd numbers is a multiple of $4$ .
[2 marks]
A2
Needed today
Expand and simplify $(3 x - 2 y)^{2}$ .
[2 marks]
Answer
A3
From chapter 1
Simplify $(- 2 x)^{3} \times (3 x)^{- 2}$ .
[2 marks]
Answer
A4
Needed today
Expand and simplify $(x + 3)^{3}$ .
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Use Pascal's triangle to expand $(3 - 4 x)^{4}$ .

Hint Index

4

, so use the row

1, 4, 6, 4, 1

. Put

(- 4 x)

in brackets every time.

B1
Use Pascal's triangle to expand $(2 - 3 x)^{5}$ .
[3 marks]
Answer

Example 2

In the expansion of $(3 + k x)^{4}$ , the coefficient of $x^{2}$ is $150$ . Find the possible values of $k$ .

Hint The

x^{2}

term is

6 \times 3^{2} \times (k x)^{2}

. Set its coefficient equal to

150

B2
In the expansion of $(2 - a x)^{5}$ , the coefficient of $x^{3}$ is $- 320$ . Find $a$ .
[3 marks]
Answer

Example 3

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

Hint Expand

(1 + 2 x)^{4}

first. An

x^{3}

comes from

2 \times

its

x^{3}

term and

- 5 x \times

its

x^{2}

term.

B3
The coefficient of $x^{2}$ in $(1 + p x) (1 - 2 x)^{4}$ is $0$ . Find $p$ .
[3 marks]
Answer

Example 4

Work out $(3 9)$ without a calculator. Hence give the $7$ th entry in the $10$ th row of Pascal's triangle.

Hint

(3 9) = \frac{9 !}{3 ! 6 !}

; cancel the

6!

. The

r

th entry in the

n

th row is

(r - 1 n - 1)

B4
Work out $(4 11)$ without a calculator. Hence give the $8$ th entry in the $12$ th row of Pascal's triangle.
[3 marks]
Answer

Practice

A-Level

C1
The $15$ th row of Pascal's triangle begins $1, 14, 91, \dots$
(a)
Find the next two numbers in the row.
[2 marks]
Answer
(b)
Hence find the coefficient of $x^{3}$ in the expansion of $(1 - 3 x)^{14}$ .
[2 marks]
Answer
Total for C1: 4 marks

C2
Show that $\frac{( n + 1 )!}{( n - 1 )!} = n (n + 1)$ . Hence solve $\frac{( n + 1 )!}{( n - 1 )!} = 132$ .
[4 marks]
Answer
C3
$(p 40) = \frac{40 !}{11 ! q !}$ . Find $q$ and the two possible values of $p$ .
[3 marks]
Answer
C4
A fair coin is thrown $12$ times. The probability of exactly $4$ heads is $(4 12) (\frac{1}{2})^{12}$ . Work out this probability, correct to $3$ significant figures, and explain why the probability of exactly $8$ heads is the same.
[3 marks]
Answer

Exam-style questions

Edexcel exam style

D1
Sam writes: "The coefficients of $(1 + 2 x)^{6}$ are the $7$ th row of Pascal's triangle, $1, 6, 15, 20, 15, 6, 1$ . So the coefficient of $x^{4}$ in $(1 + 2 x)^{6}$ is $15$ ."
Sam's answer is wrong.
(a)
Explain Sam's mistake.
[1 mark]
(b)
Find the correct coefficient of $x^{4}$ in the expansion of $(1 + 2 x)^{6}$ .
[2 marks]
Answer
(c)
The coefficient of $x^{4}$ in $(1 + 2 x)^{6}$ is equal to the coefficient of $x^{2}$ in $(1 + k x)^{6}$ , where $k$ is a positive constant. Find $k$ .
[3 marks]
Answer
Total for D1: 6 marks

D2
In this question $y$ is used for an expression in $x$ .
(a)
Use Pascal's triangle to expand $(3 + y)^{4}$ .
[2 marks]
Answer
(b)
Hence find the coefficient of $x^{2}$ in the expansion of $(3 + 2 x - x^{2})^{4}$ .
[3 marks]
Answer
(c)
Find the coefficient of $x^{3}$ in the expansion of $(3 + 2 x - x^{2})^{4}$ .
[4 marks]
Answer
Total for D2: 9 marks
D3
In this question $n$ is an integer with $n \geq 3$ .
(a)
Show that $(3 n) = \frac{n ( n - 1 ) ( n - 2 )}{6}$ .
[2 marks]
(b)
Given that $(3 n) = 4 (2 n)$ , find the value of $n$ .
[3 marks]
Answer
(c)
Hence, for this value of $n$ , find the value of the constant $k$ for which the coefficient of $x^{3}$ in $(1 + k x)^{n}$ is $7$ times the coefficient of $x^{2}$ .
[3 marks]
Answer
Total for D3: 8 marks
D4
After $4$ years, the value, in pounds, of $£2000$ invested at $r %$ per year compound interest is $2000 (1 + \frac{r}{100})^{4}$ .
(a)
Use Pascal's triangle to show that this is approximately $2000 + 80 r + A r^{2}$ , ignoring higher powers of $r$ , and find the value of the constant $A$ .
[3 marks]
Answer
(b)
Hence estimate the value of the investment after $4$ years when $r = 3$ .
[2 marks]
Answer
(c)
Work out the exact value of the investment when $r = 3$ , to the nearest penny, and explain why the estimate in part (b) is too small.
[3 marks]
Answer
Total for D4: 8 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Prove that $(r n) + (r + 1 n) = (r + 1 n + 1)$ for integers $0 \leq r < n$ .
[5 marks]
Hint 1 · What to try
Write both terms with factorials.
Hint 2 · The first line
Use the common denominator $(r + 1)! (n - r)!$ .
Hint 3 · The full method
The numerator is $n! (r + 1) + n! (n - r) = (n + 1)!$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can expand a bracket to a small power using Pascal's triangle.
I can find one coefficient, or an unknown constant from a coefficient.
I can work out n! and nCr, and use the r th entry of the n th row.
I can solve an equation with nCr or factorials in it.

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