Generalize the polynomial relationship $$(1+x)^n=\sum_{k=0}^n{n\choose k}x^k$$ for all positive integers $n$ to the situation $$(1+x+y)^n.$$
This is similar to another problem I posted where I applied the relationship to $$(a+x)^n.$$ The method there was to factor our $a^n$, but here I don't see where I can apply that logic. My suspicion is to set a third variable $z$ equal to $x+y$ and see where that takes me, but I'm not sure.