When distributing these bullets, each burglar must at least have three bullets, but no more than eight.
I have tried solving this with generating functions, but I am stuck at this part where I am not sure what type of "trick" to use to continue on..
My work (incomplete):
The generating function for this problem is $$f(x) = (x^3 + x^4 + ....+ x^8)^4$$ where we seek the coefficient of
$$=(x^3+x^4...+x^8)^4$$ $$=x^(12)( 1 + x+ x^2 + x^3 +x^4 +x^5)^4$$
and using the identities we get that the coefficient of x^12 is
(1-x^6)^4 * (1 - x)^-4
this of course is equal to :
[1-4C1(x^6) + 4C2(x^12) - 4C3(x^18) + x^23] * [-4C0 + -4C1(-x) +.......]
At this point I am totally stuck because this is a huge expansion... and the answer is 125. Is there a method I can use to quickly arrive at the answer or do I have to expand everything in?