The Binomial Expansion — Pure Mathematics 1 (9709)

Core Concept

When $n$ is a positive integer, the Binomial Theorem states that $(a + b)^n$ expands into exactly $n + 1$ terms. Each term is determined by choosing how many times $b$ is selected from the $n$ brackets in the product $(a+b)(a+b)\cdots(a+b)$.

The number of ways of choosing $r$ objects from $n$ distinct objects is denoted $^nC_r$ (read: "n choose r"), and this becomes the binomial coefficient of each term in the expansion.

Factorial notation underpins $^nC_r$: the factorial of a non-negative integer $n$, written $n!$, is the product of all positive integers up to and including $n$, with the special definition $0! = 1$.

The expansion produces terms in which the power of $a$ decreases from $n$ to $0$, and the power of $b$ increases from $0$ to $n$, with the two powers always summing to $n$.