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Find the generating function G(x) enumerates the number of options to change a dollar using an odd number of nickels, a prime number (0 and 1 excluded) of dimes, and any number of quarters.

Here's the generating function I found: $$G(x) = (x^5+x^{15}+x^{25}+\dots)(x^{20}+x^{30}+\dots)(x^{25}+x^{50}+\dots)$$

Is that look correct? It looks different from the functions that could be reduced as a closed form. In this case we're not adding infinite terms, so should I multiply them out to get the coefficient until $^{100}$? Thanks!

IGY
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    Yeah, the prime part makes it impossible to get a closed form. Quarters and nickels should have a closed form. Your quarters should include the possibility of $0$ quarters, BTW. – Thomas Andrews Jul 27 '21 at 22:44
  • @Thomas Andrews Thanks! In this case we're not adding infinite terms, so should I multiply them out to get the coefficient until $x^{100}$? – IGY Jul 27 '21 at 22:51
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    Since # of nickels required to be odd, # of quarters must also be odd. – user2661923 Jul 27 '21 at 23:36
  • @user2661923 Thanks, the highest power of each category should not be greater than 100, is that right? – IGY Jul 28 '21 at 00:02
  • Sounds right. I don't really know anything about generating functions. – user2661923 Jul 28 '21 at 00:15
  • You can cut them all off at $100$, but you don't have to. Since you're only interested in the coefficient of $x^{100}$ it won't matter if you include higher powers. There's nothing to gain by including them, though, and I would leave them out. – saulspatz Jul 28 '21 at 00:28

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