The Binomial Theorem and Pascal's Triangle

Q: How do you use Pascal's triangle to expand (a + b)^n?

Take row n (counting the top as row 0) as the coefficients. Write powers of a from n down to 0 and powers of b from 0 up to n, and multiply each pair by its coefficient.

Q: What does n choose k mean?

nk counts the ways to choose k items from n. It equals n!/k!(n-k)! and is the kth entry in row n of Pascal's triangle.

How Pascal's triangle is built, why its rows give the coefficients of $(a + b)^n$ , how to use the binomial theorem and binomial coefficients, and the link to the derivative of $x^n$ .

3 min read Supports 3 lessons

Expanding $(x + 2)^5$ by multiplying out five brackets is slow and easy to get wrong. The binomial theorem jumps straight to the answer.

Pascal's triangle

Start with a $1$ at the top. Each row begins and ends with $1$ , and every other number is the sum of the two directly above it. Point at a number below to see its two parents.

row 0

row 1

row 2

row 3

row 4

row 5

row 6

Point at a number to see the two above it that add to make it.

Counting the top row as row $0$ , row $n$ gives the coefficients of $(a + b)^n$ . Row $4$ is $1, 4, 6, 4, 1$ :

(a + b)^4 = 1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b^4

Why the triangle works

Expanding $(a + b)^4$ means choosing either $a$ or $b$ from each of the four brackets and multiplying. The coefficient of $a^2b^2$ counts how many ways you can choose $b$ from exactly two of the four brackets. There are $6$ ways.

And each row comes from the one above: multiplying $(a + b)^3$ by one more $(a + b)$ makes every term of the new row the sum of two neighbouring terms of the old row. That is exactly the triangle's addition rule.

Binomial coefficients

\binom{n}{k} = \frac{n!}{k!\,(n - k)!}

Read " $n$ choose $k$ ". $n!$ ( $n$ factorial) is $n \times (n-1) \times \dots \times 1$ . For example $\binom{4}{2} = \frac{24}{2 \cdot 2} = 6$ .

Binomial theorem

(a + b)^n = \sum_{k=0}^{n} \binom{n}{k}\, a^{\,n-k}\, b^{\,k}

The powers of $a$ count down from $n$ to $0$ while the powers of $b$ count up from $0$ to $n$ . In every term the two powers add to $n$ .

$(x + 2)^3$
Row 3 of Pascal's triangle: 1, 3, 3, 1.

1/3

Watch out

Raise the whole second term to its power. In $(x - 3)^4$ , the $b$ is $-3$ , so the signs alternate: $(-3)^2 = 9$ but $(-3)^3 = -27$ .

A link to calculus

Key idea

$(x + h)^n = x^n + nx^{n-1}h + \dots$ , where every later term contains $h^2$ or a higher power. When $h$ shrinks towards zero, those later terms vanish fastest, and that is exactly why the derivative of $x^n$ is $nx^{n-1}$ .

Common questions

How do you use Pascal's triangle to expand (a + b)^n?: Take row $n$ (counting the top as row 0) as the coefficients. Write powers of $a$ from $n$ down to $0$ and powers of $b$ from $0$ up to $n$ , and multiply each pair by its coefficient.
What does n choose k mean?: $\binom{n}{k}$ counts the ways to choose $k$ items from $n$ . It equals $\frac{n!}{k!(n-k)!}$ and is the $k$ th entry in row $n$ of Pascal's triangle.

Practise binomial theorem

The The Binomial Theorem lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

How do you use Pascal's triangle to expand (a + b)^n?

Take row $n$ (counting the top as row 0) as the coefficients. Write powers of $a$ from $n$ down to $0$ and powers of $b$ from $0$ up to $n$ , and multiply each pair by its coefficient.

What does n choose k mean?

$\binom{n}{k}$ counts the ways to choose $k$ items from $n$ . It equals $\frac{n!}{k!(n-k)!}$ and is the $k$ th entry in row $n$ of Pascal's triangle.