15. Tollbooths in Illinois accept all U.S. coins, including pennies. Carlos has

Question

15. Tollbooths in Illinois accept all U.S. coins, including pennies. Carlos has a very large supply of pennies, nickels, dimes, and quarters in his car as he drives on a toll way. He encounters a toll for $0.95 and wonders how many different ways he could use his supply of coins to pay the toll without getting change back. a) Use a generating function and computer algebra system to determine the number of ways Carlos could pay his $0.95 toll by dropping the coins to gether into the toll bin. (Assume coins of the same denomination cannot be distinguished from each other.) b) Suppose that instead of having a bin into which motorists drop the coins to pay their toll, the coins must be inserted one-by-one into a coin slot. In this scenario, Carlos wonders how many ways he could pay the 5OE95 toll when the order the coins are inserted matters. For instance, in the previous part, the use of three quarters and two dimes would be counted only one time. However, when the coins must be inserted individually into a slot, there are 10 = C(5,2) ways to insert this combination. Use an ordinary generating function and computer algebra system to determine the number of ways that Carlos could pay the $0.95 toll when considering the order the coins are inserted.

Explanation / Answer

Carlos = 1+x3+x6+…

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