I tried expanding the factorial, but I do not know how to finish the proof.
\begin{eqnarray*} \binom{~s + t~ }{s} & = & \frac{(s+t)!}{s! ~ t!}\\ & = & \frac{(s+t)(s+t-1) \cdots (t+2)(t+1)}{s!} \\ & = & \prod_{i=1}^s \frac{t + i}{i} \\ \end{eqnarray*}
How do I get $ \prod_{i=1}^s \frac{t + i}{i} = \prod_{i=1}^s \prod_{j=1}^t \frac{i + j}{i + j - 1}$?