Complexity/ polynomial time - If a problem has $x + y - 1 \choose y-1$ possible solutions & each can be evaluated in polytime, is that in P?

Question

I am trying to show the complexity of a problem and each instance's size is defined by parameters $x$ and $y$. For a given instance, I know there are $x+y-1 \choose y-1$ possible solutions. Each of these possible solutions can be evaluated in time that is polynomial in $x$ and $y$. Therefore, it seems that this problem should be solvable in polynomial time. Does this suggest my problem is polynomial time solvable? Or do I need to place further restrictions on the problem size (e.g., if $y$ is linear in $x$ then, the binomial coefficient grows as an exponential and so the logic falls through.)