Problem:
The land of Catania uses gold coins and silver coins. Gold coins are mm thick and silver coins are mm thick. In how many ways can Taylor make a stack of coins that is mm tall using any arrangement of gold and silver coins, assuming order matters?
Answer Choices:
A.
B.
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E.
Solution:
Let denote the number of gold coins and denote the number of silver coins. Then, any ordered pair of integers satisfies the equation . There are solutions to this equation: , and . For the first case, there is only way to build a tower of gold coins. For the second case, there are ways to choose where the silver coin goes, which gives ways. For the third case, there are ways to choose where the silver coins go, which gives ways. Adding up individual cases, we get that the answer is .
The problems on this page are the property of the MAA's American Mathematics Competitions