How do I prove Pascal's identity using generating functions?
$C(n,r) = C(n−1,r) + C(n−1,r−1)$ when $n$ and $r$ are positive integers with $r < n$. I am given the hint to use the identity $(1+x)^n$ = $(1+x)^{n−1}$ + $x(1+x)^{n−1}$.
So far I have simplified it down to $(1+x)^n = (1+x)^n/(x+1) + x(x+1)^n/(x+1)$. I'm not sure if I'm on the track by simplifying it like this.
What is the $x^r$ coefficient in $(1+x)^n$? What is the $x^r$ coefficient in $(1+x)^{n-1}$? What is the $x^r$ coefficient in $x(1+x)^{n-1}$? After answering these questions, take another look at the identity $(1+x)^n = (1+x)^{n-1}+x(1-x)^{n-1}$ and equate the $x^r$ coefficients on both sides.