How to work with the expectation on a products of random variables

Question

Suppose $x_i, i=1, \ldots, n$ be independent Poisson random variables with $E(x_i)=\lambda_i$.

Compute a conditional expectation with $Q, P>2$, $$ E\left[(x_1\times\ldots\times x_k)^Q(x_{k+1}\times\ldots \times x_n)^P|\sum_{i=1}^nx_i=Y\right] $$

My attempt: Denote $E\left[(x_1\times\ldots\times x_k)^Q(x_{k+1}\times\ldots \times x_n)^P|\sum_{i=1}^nx_i=Y\right]=E_{cond}\left[(x_1\times\ldots\times x_k)^Q(x_{k+1}\times\ldots \times x_n)^P\right]$

1. When $Q=P$ $$ E\left[(x_1\times\ldots\times x_n)^Q|\sum_{i=1}^nx_i=Y\right]=\left(E_{cond}(x_1 \times \ldots \times x_n)\right)^Q=\left(\frac{Y!}{(Y-n)!}\times \frac {1}{n^n}\right)^Q $$

2. When $P \neq Q$ - not sure how to split conditional expectation.