ILL 1989 Problem 25

The integers $c_{m,n}$ with $m \geq 0, \geq 0$ are defined by

$$c_{m,0} = 1 \forall m \geq 0, c_{0,n} = 1 \forall n \geq 0,$$

and

$$c_{m,n} = c_{m-1,n} - n \cdot c_{m-1,n-1} \forall m > 0, n > 0.$$

Prove that $$c_{m,n} = c_{n,m} \forall m > 0, n > 0.$$