Is $\sum\limits_{k=1}^{n-1}\binom{n}{k}x^{n-k}y^k$ always even? Is
$$ f(n,x,y)=\sum^{n-1}_{k=1}{n\choose k}x^{n-k}y^k,\qquad\qquad\forall~n>0~\text{and}~x,y\in\mathbb{Z}$$
always divisible by $2$?

Hint: Recall binomial formula
$$
(x+y)^n=\sum\limits_{k=0}^n{n\choose k} x^{n-k} y^k
$$
