The idea

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.

A full journey — read it, play with it, work it, then earn real exam marks. Everything stays on the timeline below.

In this lesson — start anywhere

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

The grind you'll never do again

Try expanding

(a + b)^4

honestly: multiply

(a+b)(a+b)

, then by

(a+b)

again, then again — four lines of algebra and a dozen chances to slip. Now imagine

(a + b)^8

. Nobody expands that by hand, because the ANSWER follows a pattern you can write straight down — and this lesson is that pattern.

The pattern

Here is the whole shape at once:

(a + b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \cdots + \binom{n}{r}a^{n-r}b^{r} + \cdots + \binom{n}{n}b^n

Keep reading — free

The rest of the explanation, plus 4 worked examples you step through move by move.

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