2008 AMC 12A Problems/Problem 19
Contents
Problem
In the expansion of what is the coefficient of ?
Solution 1
Let and . We are expanding .
Since there are terms in , there are ways to choose one term from each . The product of the selected terms is for some integer between and inclusive. For each , there is one and only one in . For example, if I choose from , then there is exactly one power of in that I can choose; in this case, it would be . Since there is only one way to choose one term from each to get a product of , there are ways to choose one term from each and one term from to get a product of . Thus the coefficient of the term is .
Solution 2
Let . Then the term from the product in question is
So we are trying to find the sum of the coefficients of minus . Since the constant term in (when expanded) is , and the sum of the coefficients of is , we find the answer to be
See Also
2008 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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All AMC 12 Problems and Solutions |