Chapter 8.4, Problem 45E

Use generating functions to prove Vandermonde’s identity: $C\left(m+n,r\right)={\displaystyle {\sum }_{k=0}^{\text{'}}C\left(m,r-k\right)C\left(n,k\right)}$. A whenever m, n, and r are nonnegative integers with r not exceeding either m or n.[Hint: Look at the coefficient of $x^r$ in both sides of ${\left(1+x\right)}^{m+n}={\left(1+x\right)}^m{\left(1+x\right)}^n$ .]