TITLE: Making Change for a Dollar (and other number partitioning problems)
QUESTION [32 upvotes]: I was trying to solve a problem similar to the "how many ways are there to make change for a dollar" problem.  I ran across a site that said I could use a generating function similar to the one quoted below:

The answer to our problem (293) is the
  coefficient of $x^{100}$ in the reciprocal
  of the following:
$(1-x)(1-x^5)(1-x^{10})(1-x^{25})(1-x^{50})(1-x^{100})$

But I must be missing something, as I can't figure out how they get from that to $293$. Any help on this would be appreciated.

REPLY [4 votes]: For a closed form formula for the question: given n cents, how many ways can I make change using pennies, nickels, dimes, quarters see either:
Making change of n cents by William Gasarch
http://arxiv.org/abs/1406.5213
This just uses recurrences.
OR
Graham-Knuth-Patashnik in their book Concrete Mathematics got a closed form using generating functions. (Actaully they did pennies,nickels,dimes,quarters,half-dollars). An online exposition of this is at
http://www.cs.umd.edu/~gasarch/BLOGPAPERS/knuthchange.pdf