Generating Functions question
Alright I have two problems both of which i THINK need to be done with a generating function.... any tips would really be appreciated (Rofl)
1)Find the number of solutions to the equation
2x + 3y + 6z = 73
where x, y, and z are non-negative integers.
I just took each case individually and got 78 possible solutions, but I was wondering if there was an easier way to do this rather than just finding each solution and counting it.
Also I am getting stuck on this one too:
How many ways are there to put 5 identical objects into n bins, where each
bin can have at most 2 objects?
2)From what we learned in class, I would guess that the generating function would be:
G(x)=(1+x+x^2)^n which is equal to [(x^3-1)/(x-1)]^n which is equal to
Then that implies that it equals ((x^3-1)^n)*SUM((a_k)x^k)
where a_k= C(k+n-1,k)
Usually I would just plug in 5 for k, and get my answer, but what do I do with the (x^3-1)^n??
Any help would be great!!!!