The binomial expansion

Expanding (a + b)^n with binomial coefficients — Pascal's triangle and nCr — and picking out a single term or coefficient without expanding everything.

Pure · Sequences & series

The binomial expansion

Expanding (a + b)^n with binomial coefficients — Pascal's triangle and nCr — and picking out a single term or coefficient without expanding everything.

Why it works

Expanding

(a + b)^n

by hand is just repeated multiplication, but the pattern of the answer is predictable, which lets you skip the grind.

(a + b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \dots + \binom{n}{n}b^n.

Reading along a row: the power of

a

falls from

n

0

while the power of

b

rises from

0

n

— the two powers always add to $n$ — and each term carries a binomial coefficient

\binom{n}{r}

. Those coefficients are the rows of Pascal's triangle, or directly

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

Two things people drop:

Raise the whole term. In $(2 + 3x)^3$ , a term like $\binom{3}{2}(2)^1(3x)^2$

needs

(3x)^2 = 9x^2

, not

3x^2

. The coefficient and the variable get the power.

The constant $a$ is raised too. $(2 + 3x)^n$ is not $(1 + 3x)^n$ — the powers

2

matter.

Picking one term. You rarely need the whole expansion. The term in

b^r

\binom{n}{r}\,a^{n-r}\,b^r,

so to find, say, the coefficient of

x^3

you set

r = 3

and evaluate that one term. (Mind signs: in

(a - bx)^n

the term carries

(-1)^r