The problem
Let $X \sim Binomial(l, p)$ and $\sum_{i=1}^{m} X_i = n$, where $l$, $m$, $n$ and $p$ are all known constants. Find: $$Var\left(\sum_{i=1}^{m} X_i^2 \right)$$
What I tried
My first attempt was using $Var\left(\sum_{i=1}^{m} X_i^2 \right) = \sum_{i=1}^{m} Var(X_i^2)$, but this only holds when $X_i$ are independent. As $\sum_{i=1}^{m} X_i = n$, the $X_i$ are clearly dependent, so I think I need to use the covariance between $X_i$ and $X_j$ like this:
$$ Var\left(\sum_{i=1}^{m} X_i^2 \right) = \sum_{i=1}^{m} Var(X_i^2) + \sum_{i\neq j} Cov(X_i^2, X_j^2) $$
However, I have no clue how I can calculate the covariance while taking into account $\sum_{i=1}^{m} X_i = n$. Can anyone help me out?
Bonus points if you also know how to do it for the case where $X_i \sim Trinomial(l, p, q)$ :)