Putnam 1962 B1

Let $x^{(n)}=x(x-1)\cdots (x-n+1)$ for $n$ a positive integer and let $x^{(0)}=1.$ Prove that $$(x+y)^{(n)}= \sum_{k=0}^{n} \binom{n}{k} x^{(k)} y^{(n-k)}.$$