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Worksheet 8.1 Intro to Ordinary Generating Functions
Activity 8.1.1 .
Our goal is to use generating functions to find the number of ways to make certain amounts of money for using coins.
(a) If you want to find the number of ways to make $1, you should plan to look for the coefficient on
\(x^{n}\) for what
\(n\text{?}\)
(b) Suppose first that you only have pennies and nickels. What’s the generating function? How many ways to make $1 using pennies and nickels?
(c) Now suppose you have pennies, nickels, dimes, and quarters. What’s the generating function? Number of ways to make $1? Number of ways to make $0.83?
(d) Suppose you have pennies, nickels, dimes, quarters, half dollars, and dollar coins. What’s the generating function? What’s the number of ways to make $1? What’s the number of ways to make $2.50?
