Binomial coefficient

Definition

Definition

Let $n$ and $p$ be two natural integers, we call binomial coefficient, and we note $\binom{n}{p}$, the following number

$$\binom{n}{p} = \frac{n!}{p!(n-p)!}$$

Remark

For all $n \in \mathbb N$, we have $\binom{n}{0} = \binom{n}{n} = 1$

Results

Proposition

For all $(n, p) \in \mathbb N^2$ such that $n \geq p \geq 1$, we have

$$\binom{n}{p} = \frac{n}{p} \binom{n-1}{p-1} = \frac{n-p+1}{p}\binom{n}{p-1}$$

Proof

It is straightforward from the definition by putting $p$ in the denominator and in the numerator the first or the last term.

Proposition

Let $n$ and $p$ be two natural integers, we have

$$\begin{align} \binom{n}{p} = & \binom{n}{n-p} \quad \text{if} \quad p \leq n. \\ \binom{n}{p} = & \binom{n-1}{p} + \binom{n-1}{p-1} \quad \text{if} \quad p \geq 1 \end{align}$$

Proof

By calculation, from the definition of $\binom{n}{p}$.