Pascal's Rule

Theorem
Let $\dbinom n k$ be a binomial coefficient.

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

This is also valid for the real number definition:


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

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


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

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

Also known as
Some sources give this as Pascal's identity.

Also presented as
Some sources present this as:


 * $\dbinom n k + \dbinom n {k + 1} = \dbinom {n + 1} {k + 1}$

Also see

 * Definition:Pascal's Triangle