Let $x_i, i=1, ..., n$ be identically distributed $(m, p)$-multivariate random variables, such that $\sum_{i=1}^{n-1}x_i=y$ and $y+x_n=m$. Let $k_i, i=1,..., n$ is in $N$ and such that $\sum_i{k_i}=K$.
Find $$ E\left[\left(\prod_{i=1}^{n-1}x_i^{k_i}\right)(m-y)^{K-\sum_{i=1}^{n-1}k_i}\right]. $$
I have been trying to open up brackets inside of the expectation, but the computations does not seems get easier.
This is a slightly weird exercise; there's just a bit of algebraic unpacking to do, without any probabilistic content.
With $\sum_ik_i=K$, the exponent is $K-\sum_{i=1}^{n-1}k_i=k_n$. With $y+x_n=m$, the base is $m-y=x_n$. So the last factor is just the factor $x_n^{k_n}$ that’s missing in the product, so the whole thing is just
$$ E\left[\prod_{i=1}^nx_i^{k_i}\right]\;, $$
which would have been
$$ \prod_{i=1}^nE\left[x_i^{k_i}\right] $$
if you hadn’t removed the independence assumption. I don’t think there’s more to say about it than that without knowing anything about the distribution.