Probability density in terms of expansions

I am studying the problem of angular distributions.

Let's say I have a two-electron wavefucntion $\psi(r_1,r_2,t)$ from which I wish to find the distribution of electrons at some time $t$.

This wavefunction is expanded over the complete set of two-electron, time-independent, definite energy and angular momentum eigenvectors as follows:

$$\psi(r_1,r_2,t) = \sum_{nL}C_{nL}(t)\phi_{nL}(r_1,r_2)$$

where $C_{nL}(t)$ is a complex time dependent expansion co-efficient.

Now suppose I want to do the following with this expansion: $$\psi^*(r_1,r_2,t)\psi(r_1,r_2,t) = |\psi(r_1,r_2,t)|^2 .$$

This will give me the probability density for finding the two electrons at position $r_1$ and $r_2$ at time $t$.

In mathematical terms, I need to find the product of the two expansions,

$$\psi(r_1,r_2,t) = \sum_{nL}C_{nL}(t)\phi_{nL}(r_1,r_2)$$ $$\psi^*(r_1,r_2,t) = \sum_{nL}C_{nL}^*(t)\phi^*_{nL}(r_1,r_2)$$

My problem arises from the fact that I am completely inexperienced with summations. How do I multiply these two expansions? Do I have to add a prime to the $n$ and $L$ on one of the expansions (e.g $n \rightarrow n'$) and if so, could you provide and explanation as to why this is necessary?

Also, is a product of two sums over two different indexes the same as sum over all indexes of the products of the insides of the sums? Something like this?

$$\psi^*(r_1,r_2,t)\psi(r_1,r_2,t) = \sum_{nL}\sum_{n'L'}C_{n'L'}^*(t)C_{nL}(t)\phi^*_{n'L'}(r_1,r_2)\phi_{nL}(r_1,r_2) \;\;.$$

I'm just guessing this is the form. It'd be great if someone explained why this is the form if it's correct (i.e explain the need for the primes), or why this is wrong.

Indices are just labels. Why don't you write down 2 sums with only, say, three terms in, $$S_1 = \sum_{i=1}^3 x_i = x_1 + x_2 + x_3$$ and $$S_2 = \sum_{i=1}^3 y_i = y_1 +y_2 +y_3$$, and have a play with the product $$S_1S_2$$ to see what happens.