Balls and Bins Varaince

Question

I am trying to self-learn some statistic concepts and came across this apparently famous "Balls and Bins" Question, but I couldn't wrap my head around it.

So suppose we have 5 balls and 12 bins and *each ball is * equally likely to land in any bin.

Let there be some random variable $X_{i}$ denoting the number of balls in bin i.

We have to find the variance of $X_{1}+X_{2}...+X_{12}$

My half baked take is that we can try finding the expected value of $X_{i}$ and go from there. I assume, probably incorrectly, that the expected load on each bin would be $$\frac{5}{12}$$

I don't know if my computation is correct, or where to go from here even if it is correct.

Answer 1

It's a trick question. The sum

$$ X_1 + X_2 + \cdots + X_{12} $$

is always exactly 5, for every outcome. So the variance is zero.