My question is as follows:
(a) 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?
(b) Suppose each student who is a candidate votes for herself or himself. Now what is the generating function and the required coefficient?
(c) Suppose no candidate receives a majority of the vote. Repeat part (a).
I found the first two parts relatively easy, and got answers of $(1 + x + x^2 + ... + x^{25})^4$, looking for the coefficient of $x^{25}$ and $(1 + x + x^2 + ... + x^{21})^4$, looking for the coefficient of $x^{21}$ respectively.
However I am having trouble with part c). My logic and attempt is as follows:
A majority of 25 is 13, so in order to prevent this, each candidate will have 12 or less votes, making the generating function:
$$(1 + x + x^2 + ... + x^{12})^4$$
and we would be looking for the coefficient of $x^{25}$ again to find the answer.
Is this correct, or am I missing something here? Thanks.