I'm working on a problem that needs to be solved using EM algorithm. In doing that, I have to evaluate an expectation that I actually have no idea how to.

Consider: $Y$ as a a fixed observed integer and consider $Z$ as a random variable which has a Poisson distribution with rate $\theta$:

$$ Z \sim \operatorname{Pois}(\theta)$$
$$ Z|Y \sim \operatorname{Binom}(n = y, p)$$

How can I compute the expectation below with respect to $Z$?

$$E\left(\frac{y!}{Z! (y - Z)!}\right)$$