In the case of discrete random variables, the convolution is obtained by summing a series of products of the probability mass functions (pmfs) of the two variables. But what if we have 50 variables? What is the approach in this case?
What is the general formula to calculate the convolution of more than 2 probability distributions?
The general approach is to iterate over each of the individual distributions. For example, to get the distribution of $W = X + Y + Z$, you would do:
$$\begin{eqnarray}p_W (w) & = & P(W = w) \\ & = & P(X + Y + Z = w) \\ & = & \sum_x P(X = x \land Y + Z = w - x) \\ & = & \sum_x P(Y + Z = w - x | X = x) P(X = x) \\ & = & \sum_x \sum_y P(Y = y \land Z = w - x - y | X = x) P(X = x) \\ & = & \sum_x \sum_y P(Z = w - x - y | X = x \land Y = y) P(Y = y | X = x) P(X = x) \\ & = & \sum_x \sum_y p_X (x) p_{Y|X = x} (y) p_{Z|X = x \land Y = y} (w - x - y) \\ & = & \sum_x \sum_y p_X (x) p_Y (y) p_Z(w - x - y) \mbox{ if } X, Y, Z \mbox{ are independent} \end{eqnarray}$$
If you have fifty random variables, you take every combination of ways those 50 variables can take values that sum to some total $T$, you find the probability of that combination of values, and you add it all together.
There are some shortcuts - some distributions add "nicely", e.g. independent Poisson distributions add to another Poisson. You can also use generating functions to simplify the process - if you set $f_X (x) = \sum_i p_X(i) x^i$, i.e. form a power series whose coefficients are the probabilities of each outcome, then $f_{X+Y} (z) = f_X (z) f_Y(z)$, and if you can find a familiar form for $f_X$ and $f_Y$ then you can potentially also work out $f_{X+Y}$ quickly.