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The following coins were in circulation in the United States in 1875: Indian-head penny, a bronze two-cent piece, a silver three-cent piece, a nickel three-cent piece, the shield nickel (worth five-cents), the seated liberty half-dime, dime, twenty-cent piece, quarter, half-dollar, and silver dollar , and the Indian-head gold dollar.

How many ways were there to make twenty cents in change? How about twenty-five cents?

Write down a generating function in the form of a rational function for the number of ways to make k cents in change.

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    $\begingroup$ If worse comes to worst, $20$ and $25$ are small enough that you can answer the first part by brute force. For the second part, could you write down a generating function if you had only two-cent pieces available? What if you had pennies and two-cent pieces? $\endgroup$ – Brian M. Scott Feb 17 '15 at 1:54
  • $\begingroup$ Is this a question for a class? From a textbook? $\endgroup$ – Tad Feb 17 '15 at 2:27

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