I am working on my combinatorics hw and there's something I am not quite sure:
Use a generating function for modeling the number of different election outcomes in an election for class president if 25 students are voting among four candidates. Which coefficient do we want?
I know how to find the generating function, but I am not sure about which coefficient we want. for generating function, I got
$$g(x) = (1+x+x^2+\ldots+x^{25})^4$$
So if we expand the equation, we'd get a bunch of coefficients, but I am not sure which want is what we want and why.
Thanks in advance!
One way to look at this problem is to see that it is like assigning $25$ balls to $4$ different bins without restriction. The number of ways to do this is $$\binom{25+4-1}{25} = 3276$$ If we wanted to use a generating function, we have that a "box" (candidate) gets filled with either $0, 1, ... 25$ "balls" (votes) so we have $$f(x) = (1+x+x^2+...+x^{25})^4$$ The answer is the coefficient of $x^{25}$ since we have $25$ votes in total. This gives the same answer as before, $3276$
Expanded function on WolframAlpha