Chapter 8.4, Problem 44E

Use generating functions to prove Pascal’s identity: $C\left(n,r\right)=C\left(n-1,r\right)+C\left(n-1,r-1\right)$ when n and r are positive integers with $r<n$ .[Hint: Use the identity

${\left(1+x\right)}^n={\left(1+x\right)}^{n-1}+x{\left(1+x\right)}^{n-1}$ .]