On the MathWorld site, it gives a formula for calculating the probability of rolling a total of **p** on **n** dice with **s** sides each. It starts by saying the number of possible ways to roll this total is the coefficient of the x^p term in the multinomial:

(x + x^2 + … + x^s)^n

The rest of the article is simply about how to calculate this multinomial coefficient using a number of binomial coefficients. But it doesn’t elaborate on why the multinomial coefficient itself is the correct answer, treating it as self-evident. However, it doesn’t seem obvious to me. How can we prove that this is the case?