Sec 3 A-Math · Binomial theorem

ⁿCᵣ questions with answers

Practice questions on ⁿcᵣ, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Without using a calculator, find the value of $\,{}^{8}C_{2}$ .

Step 1 of 2

ⁿCᵣ = r numbers counting down from n, over r!

\,{}^{8}C_{2} = \dfrac{8 \times 7}{2 \times 1}

Try a question: ⁿCᵣ

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.

Questions

Question 1Work out ⁿCᵣ
Without using a calculator, find the value of $\,{}^{7}C_{5}$ .
Show the worked solution
1. Step 1: ⁿCᵣ = ⁿCₙ₋ᵣ: use the smaller of r and n − r $\,{}^{7}C_{5} = {}^{7}C_{2}$
2. Step 2: ⁿCᵣ = r numbers counting down from n, over r! $\,{}^{7}C_{2} = \dfrac{7 \times 6}{2 \times 1}$
3. Step 3: Work out its value $= 21$
Question 2Work out ⁿCᵣ
Without using a calculator, find the value of $\,{}^{7}C_{2}$ .
Show the worked solution
1. Step 1: ⁿCᵣ = r numbers counting down from n, over r! $\,{}^{7}C_{2} = \dfrac{7 \times 6}{2 \times 1}$
2. Step 2: Work out its value $= 21$
Question 3Find n from ⁿC₂
Given that $\,{}^{n}C_{2} = 105$ , find the value of $n$ .
Show the worked solution
1. Step 1: Write ⁿC₂ in terms of n, and set it equal to the value given $\dfrac{n(n - 1)}{2} = 105$
2. Step 2: Multiply both sides by 2, then make one side zero $n(n - 1) = 210$ $n^2 - n - 210 = 0$
3. Step 3: Factorise and solve: n must be a positive whole number $(n - 15)(n + 14) = 0$ $n = 15$ ( $n = -14$ is rejected: n can't be negative.)
Question 4Find n from ⁿC₂
Given that $\,{}^{n}C_{2} = 210$ , find the value of $n$ .
Show the worked solution
1. Step 1: Write ⁿC₂ in terms of n, and set it equal to the value given $\dfrac{n(n - 1)}{2} = 210$
2. Step 2: Multiply both sides by 2, then make one side zero $n(n - 1) = 420$ $n^2 - n - 420 = 0$
3. Step 3: Factorise and solve: n must be a positive whole number $(n - 21)(n + 20) = 0$ $n = 21$ ( $n = -20$ is rejected: n can't be negative.)
Question 5Pascal's triangle
Use Pascal's triangle to find the coefficient of $x^{4}$ in the expansion of $(1 + x)^{5}$ .
Show the worked solution
1. Step 1: Pascal's triangle: each row starts and ends with 1, and every other number is the sum of the two above it $\def\arraystretch{2}\begin{array}{c} \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 2\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 3\kern0.5em 3\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 4\kern0.5em 6\kern0.5em 4\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 5\kern0.5em 10\kern0.5em 10\kern0.5em 5\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \end{array}$
2. Step 2: Row 5 (the top row is row 0) gives the coefficients of 1, x, x², …: count along from 0 $\text{Coefficient of } x^{4} = 5$
Question 6Pascal's triangle
Use Pascal's triangle to find the value of $\,{}^{5}C_{3}$ .
Show the worked solution
1. Step 1: Pascal's triangle: each row starts and ends with 1, and every other number is the sum of the two above it $\def\arraystretch{2}\begin{array}{c} \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 2\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 3\kern0.5em 3\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 4\kern0.5em 6\kern0.5em 4\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 5\kern0.5em 10\kern0.5em 10\kern0.5em 5\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \end{array}$
2. Step 2: Row 5 (the top row is row 0) gives ⁿC₀, ⁿC₁, ⁿC₂, …: count along from 0 $\,{}^{5}C_{3} = 10$

More on binomial theorem

Notes: Binomial theorem Worksheet with answers (PDF)

Endless questions on ⁿcᵣ

A free account gives you new questions every time, step-by-step help, and tracks what you've mastered.

Start free

Questions

Question 1Work out ⁿCᵣ

Without using a calculator, find the value of $\,{}^{7}C_{5}$ .

Show the worked solution

Step 1: ⁿCᵣ = ⁿCₙ₋ᵣ: use the smaller of r and n − r $\,{}^{7}C_{5} = {}^{7}C_{2}$
Step 2: ⁿCᵣ = r numbers counting down from n, over r! $\,{}^{7}C_{2} = \dfrac{7 \times 6}{2 \times 1}$
Step 3: Work out its value $= 21$

Question 2Work out ⁿCᵣ

Without using a calculator, find the value of $\,{}^{7}C_{2}$ .

Show the worked solution

Step 1: ⁿCᵣ = r numbers counting down from n, over r! $\,{}^{7}C_{2} = \dfrac{7 \times 6}{2 \times 1}$
Step 2: Work out its value $= 21$

Question 3Find n from ⁿC₂

Given that $\,{}^{n}C_{2} = 105$ , find the value of $n$ .

Show the worked solution

Step 1: Write ⁿC₂ in terms of n, and set it equal to the value given $\dfrac{n(n - 1)}{2} = 105$
Step 2: Multiply both sides by 2, then make one side zero $n(n - 1) = 210$ $n^2 - n - 210 = 0$
Step 3: Factorise and solve: n must be a positive whole number $(n - 15)(n + 14) = 0$ $n = 15$ ( $n = -14$ is rejected: n can't be negative.)

Question 4Find n from ⁿC₂

Given that $\,{}^{n}C_{2} = 210$ , find the value of $n$ .

Show the worked solution

Step 1: Write ⁿC₂ in terms of n, and set it equal to the value given $\dfrac{n(n - 1)}{2} = 210$
Step 2: Multiply both sides by 2, then make one side zero $n(n - 1) = 420$ $n^2 - n - 420 = 0$
Step 3: Factorise and solve: n must be a positive whole number $(n - 21)(n + 20) = 0$ $n = 21$ ( $n = -20$ is rejected: n can't be negative.)

Question 5Pascal's triangle

Use Pascal's triangle to find the coefficient of $x^{4}$ in the expansion of $(1 + x)^{5}$ .

Show the worked solution

Step 1: Pascal's triangle: each row starts and ends with 1, and every other number is the sum of the two above it $\def\arraystretch{2}\begin{array}{c} \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 2\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 3\kern0.5em 3\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 4\kern0.5em 6\kern0.5em 4\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 5\kern0.5em 10\kern0.5em 10\kern0.5em 5\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \end{array}$
Step 2: Row 5 (the top row is row 0) gives the coefficients of 1, x, x², …: count along from 0 $\text{Coefficient of } x^{4} = 5$

Question 6Pascal's triangle

Use Pascal's triangle to find the value of $\,{}^{5}C_{3}$ .

Show the worked solution

Step 1: Pascal's triangle: each row starts and ends with 1, and every other number is the sum of the two above it $\def\arraystretch{2}\begin{array}{c} \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 2\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 3\kern0.5em 3\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 4\kern0.5em 6\kern0.5em 4\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \\ \kern1.05em\mathclap{1}\kern1.05em\kern0.5em 5\kern0.5em 10\kern0.5em 10\kern0.5em 5\kern0.5em \kern1.05em\mathclap{1}\kern1.05em \end{array}$
Step 2: Row 5 (the top row is row 0) gives ⁿC₀, ⁿC₁, ⁿC₂, …: count along from 0 $\,{}^{5}C_{3} = 10$