Definition:Binomial Coefficient

Let $$m, n \in \N$$.

Then the symbol $$\binom n m$$ is interpreted as:



\binom n m = \begin{cases} \displaystyle \frac {n!} {m! \left({n - m}\right)!} & : m \le n \\ 0 & : m > n \end{cases} $$

The number $$\binom n m$$ is known as a binomial coefficient.

See the Binomial Theorem for the reason why.

$$\binom n m$$ is read n choose m.

Recursive Definition
The binomial coefficients can be defined using the following recurrence relation:


 * $$\binom n k = \begin{cases}

1 & : k = 0 \\ 0 & : k > n \\ \binom{n-1}{k-1} + \binom{n-1}{k} & : \text{otherwise} \end{cases}$$

This relation is known as Pascal's Rule.