$\#P$ closure under binomial coefficients?
Let me take a guess: you mean an operation of the form $f(x) \mapsto \binom{f(x)}{d}$, where $d$ is a constant. Let's assume also that $f(x)$ is the number of $y$ such that $\phi(x,y)$. Then $\binom{f(x)}{d}$ is the number of $d$-tuples $y_1,\ldots,y_d$ such that $y_1 &lt; \cdots &lt; y_d$ and $\phi(x,y_1) \land \cdots \land \phi(x,y_d)$.