Using a generating function for piggy bank problem A piggy bank contains 45 loonies and 25 toonies. How
many ways can the coins be divided so that Jamie gets no loonies, Julie gets no toonies but
at least 10 loonies, and Brenda gets an odd number of toonies? Use generating functions to
solve the problem. (Note that you are only interested in how many of each type of coin each
sibling gets, not which specific coins they get).
So far, my only idea is to split the problem up into 2 generating functions, and then multiply the two answers:
For the loonies,
$g(x)=x^{10}(1+x+\cdots+x^{35})(1+x+x^2+\cdots+x^{45})
= x^{10}\frac{(x^{36}-1)}{x-1}\frac{(x^{46}-1)}{x-1}$
And I would do similarly for the toonies. My problem is that I'm not sure if this is even a way to do it, and my biggest problem is that I still don't understand how to find the coefficient of $x^{45}$ in the generating function I have shown, despite reading through all of my notes. 
Thanks very much in advance for any help!!

To answer your first question, your method is essentially correct. If you have $a$ ways of distributing the loonies and $b$ ways of distributing the toonies, then you have $ab$ ways of doing both. See also the rule of product and this video.
To answer your biggest question, you know that each loony will go to one of two people, and you know that 10 of unlabeled loonies have a predetermined destination, so the question is how can the other 35 be distributed into two groups? You want to know how many ways there are to write 35 as the sum of two nonnegative integers. Well, there's $0+35$, and $1+34$... surely you can count them all.
More generally, with a generating function you can take the Taylor series expansion centered at 0 to find the coefficient you want, but that's not really reasonable to do by hand with the 45th coefficient.
