Evaluate the sum $\sum_{n=1}^{b}n\binom{a}{n}\binom{b}{n}$ Let $a$ and $b$ be natural numbers such that $a \geq b \geq 1$. How can we evaluate the following sum?
$$ \sum_{n=1}^{b}n\binom{a}{n}\binom{b}{n}$$

$$\begin{align}
\sum_{n=1}^b n\binom an\binom bn
&=\sum_{n=1}^b \binom an\cdot n\binom bn\\
&=\sum_{n=1}^b \binom an\cdot b\binom {b-1}{n-1}\\
&=b\sum_{n=1}^b \binom a{a-n}\binom {b-1}{n-1}\\
&=b\binom {a+b-1}{a-1}
&&\text{(Vandermonde)}\\
&=b\binom {a+b-1}{b}\\
&=(a+b-1)\binom {a+b-2}{b-1} \\
&=(a+b-1)\binom {a+b-2}{a-1}\end{align}$$
