Sec 3 · Additional Mathematics

Binomial theorem

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1. Combinations

Name	Formula	Example
${}^{n}C_{r}$	$\dfrac{n!}{r!\,(n - r)!}$	${}^{8}C_{3} = 56$
	${}^{n}C_{0} = 1$	${}^{8}C_{0} = 1$
	${}^{n}C_{n} = 1$	${}^{8}C_{8} = 1$
	${}^{n}C_{1} = n$	${}^{8}C_{1} = 8$
	${}^{n}C_{2} = \dfrac{n(n - 1)}{2}$	${}^{8}C_{2} = 28$
symmetry	${}^{n}C_{r} = {}^{n}C_{n - r}$	${}^{8}C_{5} = {}^{8}C_{3}$

Pascal's triangle

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Row $n$ (the top row is row $0$ ) gives ${}^{n}C_{0}, {}^{n}C_{1}, \ldots, {}^{n}C_{n}$
Each row starts and ends with $1$ ; every other number is the sum of the two above it

📖 Worked example

Given that $\,{}^{n}C_{2} = 28$ , find the value of $n$ .

Step 1 of 3

Write ⁿC₂ in terms of n, and set it equal to the value given

\dfrac{n(n - 1)}{2} = 28

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2. Binomial expansions

Name	Formula
expansion	$(a + b)^{n} = a^{n} + {}^{n}C_{1}a^{n - 1}b + {}^{n}C_{2}a^{n - 2}b^{2} + \ldots + b^{n}$ ( $n$ a positive integer; ${}^{n}C_{r}$ is also written $\binom{n}{r}$ )
general term	$T_{r + 1} = {}^{n}C_{r}\,a^{n - r}b^{r}$

First three terms, ascending powers of x

\displaystyle (1 + 2x)^{5} = 1 + 10x + 40x^{2} + \ldots

${}^{5}C_{1}(2x)$ , which is $10x$
${}^{5}C_{2}(2x)^{2}$ , which is $40x^{2}$

Both terms in the bracket raised to powers

\displaystyle (2 - x)^{6} = 64 - 192x + 240x^{2} + \ldots

${}^{6}C_{1}(2)^{5}(-x)$ , which is $-192x$
${}^{6}C_{2}(2)^{4}(-x)^{2}$ , which is $240x^{2}$

📖 Worked example

Find the first three terms in the expansion of $(2 - 3x)^{6}$ in ascending powers of $x$ .

Step 1 of 3

(a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + …: write the terms out

(2 - 3x)^{6} = {}^{6}C_{0}(2)^{6} + {}^{6}C_{1}(2)^{5}(-3x) + {}^{6}C_{2}(2)^{4}(-3x)^{2} + \ldots

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3. A particular term or coefficient

Term independent of x: the power of x is 0

\displaystyle \left(x - \dfrac{1}{x}\right)^{8}

$T_{r + 1} = {}^{8}C_{r}\,x^{8 - r}\left(-\dfrac{1}{x}\right)^{r}$
power of $x$ : $8 - 2r = 0$ , so $r = 4$
term: ${}^{8}C_{4}(-1)^{4}$ , which is $70$

Coefficient of x in a product

\displaystyle (1 - 3x)(1 - x)^{4} = (1 - 3x)(1 - 4x + \ldots)

$x$ terms: $1(-4x)$ and $(-3x)(1)$
coefficient: $-4 - 3$ , which is $-7$

📖 Worked example

Find the term independent of $x$ in the expansion of $\left(x - \dfrac{1}{x}\right)^{8}$ .

Step 1 of 3

Write the general term: Tᵣ₊₁ = ⁿCᵣ (first term)ⁿ⁻ʳ (second term)ʳ, then collect the powers of x

T_{r+1} = {}^{8}C_{r}(x)^{8 - r}\left(\dfrac{-1}{x}\right)^{r}

= {}^{8}C_{r}(-1)^{r} x^{8 - 2r}

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4. Finding an unknown

In $(1 + ax)^{n}$	Coefficient
$x$	$na$
$x^{2}$	${}^{n}C_{2}a^{2}$ , so $\dfrac{n(n - 1)}{2}a^{2}$

(1 + ax)⁶: the coefficient of x is 18

\displaystyle 6a = 18

e.g.

\displaystyle 6a = 18

\displaystyle a = 3

📖 Worked example

In the expansion of $(1 + 2x)^{n}$ , the coefficient of $x^2$ is $60$ . Find the value of $n$ .

Step 1 of 3

Write the x² term, then write ⁿC₂ in terms of n

T_{3} = {}^{n}C_{2}(2x)^{2}

= \dfrac{n(n - 1)}{2} \times 4x^2

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Sec 3 · Additional Mathematics

Binomial theorem

Practise binomial theorem with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Combinations

Name	Formula	Example
${}^{n}C_{r}$	$\dfrac{n!}{r!\,(n - r)!}$	${}^{8}C_{3} = 56$
	${}^{n}C_{0} = 1$	${}^{8}C_{0} = 1$
	${}^{n}C_{n} = 1$	${}^{8}C_{8} = 1$
	${}^{n}C_{1} = n$	${}^{8}C_{1} = 8$
	${}^{n}C_{2} = \dfrac{n(n - 1)}{2}$	${}^{8}C_{2} = 28$
symmetry	${}^{n}C_{r} = {}^{n}C_{n - r}$	${}^{8}C_{5} = {}^{8}C_{3}$

Pascal's triangle

\displaystyle \def\arraystretch{1.5}\begin{array}{c} \mathclap{1} \\ \mathclap{1}\kern1.7em \mathclap{1} \\ \mathclap{1}\kern1.7em \mathclap{2}\kern1.7em \mathclap{1} \\ \mathclap{1}\kern1.7em \mathclap{3}\kern1.7em \mathclap{3}\kern1.7em \mathclap{1} \\ \mathclap{1}\kern1.7em \mathclap{4}\kern1.7em \mathclap{6}\kern1.7em \mathclap{4}\kern1.7em \mathclap{1} \\ \mathclap{1}\kern1.7em \mathclap{5}\kern1.7em \mathclap{10}\kern1.7em \mathclap{10}\kern1.7em \mathclap{5}\kern1.7em \mathclap{1} \\ \mathclap{1}\kern1.7em \mathclap{6}\kern1.7em \mathclap{15}\kern1.7em \mathclap{20}\kern1.7em \mathclap{15}\kern1.7em \mathclap{6}\kern1.7em \mathclap{1} \end{array}

Row $n$ (the top row is row $0$ ) gives ${}^{n}C_{0}, {}^{n}C_{1}, \ldots, {}^{n}C_{n}$
Each row starts and ends with $1$ ; every other number is the sum of the two above it

📖 Worked example

Given that $\,{}^{n}C_{2} = 28$ , find the value of $n$ .

Step 1 of 3

Write ⁿC₂ in terms of n, and set it equal to the value given

\dfrac{n(n - 1)}{2} = 28

Practise this →

2. Binomial expansions

Name	Formula
expansion	$(a + b)^{n} = a^{n} + {}^{n}C_{1}a^{n - 1}b + {}^{n}C_{2}a^{n - 2}b^{2} + \ldots + b^{n}$ ( $n$ a positive integer; ${}^{n}C_{r}$ is also written $\binom{n}{r}$ )
general term	$T_{r + 1} = {}^{n}C_{r}\,a^{n - r}b^{r}$

First three terms, ascending powers of x

\displaystyle (1 + 2x)^{5} = 1 + 10x + 40x^{2} + \ldots

${}^{5}C_{1}(2x)$ , which is $10x$
${}^{5}C_{2}(2x)^{2}$ , which is $40x^{2}$

Both terms in the bracket raised to powers

\displaystyle (2 - x)^{6} = 64 - 192x + 240x^{2} + \ldots

${}^{6}C_{1}(2)^{5}(-x)$ , which is $-192x$
${}^{6}C_{2}(2)^{4}(-x)^{2}$ , which is $240x^{2}$

📖 Worked example

Find the first three terms in the expansion of $(2 - 3x)^{6}$ in ascending powers of $x$ .

Step 1 of 3

(a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + …: write the terms out

(2 - 3x)^{6} = {}^{6}C_{0}(2)^{6} + {}^{6}C_{1}(2)^{5}(-3x) + {}^{6}C_{2}(2)^{4}(-3x)^{2} + \ldots

Practise this →

3. A particular term or coefficient

Term independent of x: the power of x is 0

\displaystyle \left(x - \dfrac{1}{x}\right)^{8}

$T_{r + 1} = {}^{8}C_{r}\,x^{8 - r}\left(-\dfrac{1}{x}\right)^{r}$
power of $x$ : $8 - 2r = 0$ , so $r = 4$
term: ${}^{8}C_{4}(-1)^{4}$ , which is $70$

Coefficient of x in a product

\displaystyle (1 - 3x)(1 - x)^{4} = (1 - 3x)(1 - 4x + \ldots)

$x$ terms: $1(-4x)$ and $(-3x)(1)$
coefficient: $-4 - 3$ , which is $-7$

📖 Worked example

Find the term independent of $x$ in the expansion of $\left(x - \dfrac{1}{x}\right)^{8}$ .

Step 1 of 3

Write the general term: Tᵣ₊₁ = ⁿCᵣ (first term)ⁿ⁻ʳ (second term)ʳ, then collect the powers of x

T_{r+1} = {}^{8}C_{r}(x)^{8 - r}\left(\dfrac{-1}{x}\right)^{r}

= {}^{8}C_{r}(-1)^{r} x^{8 - 2r}

Practise this →

4. Finding an unknown

In $(1 + ax)^{n}$	Coefficient
$x$	$na$
$x^{2}$	${}^{n}C_{2}a^{2}$ , so $\dfrac{n(n - 1)}{2}a^{2}$

(1 + ax)⁶: the coefficient of x is 18

\displaystyle 6a = 18

e.g.

\displaystyle 6a = 18

\displaystyle a = 3

📖 Worked example

In the expansion of $(1 + 2x)^{n}$ , the coefficient of $x^2$ is $60$ . Find the value of $n$ .

Step 1 of 3

Write the x² term, then write ⁿC₂ in terms of n

T_{3} = {}^{n}C_{2}(2x)^{2}

= \dfrac{n(n - 1)}{2} \times 4x^2

Practise this →

Try a question: Binomial theorem

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise binomial theorem with step-by-step help

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