Pascal's Rule

Theorem
Let $\displaystyle \binom n k$ be a binomial coefficient.

For positive integers $n, k$ with $1 \le k \le n$:
 * $\displaystyle \binom n {k-1} + \binom n k = \binom {n+1} k$

This is also valid for the real number definition:


 * $\displaystyle \forall r \in \R, k \in \Z: \binom r {k-1} + \binom r k = \binom {r+1} k$

Thus the binomial coefficients can be defined using the following recurrence relation:


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

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

Also see

 * Pascal's Triangle