2017 AIME I Problems/Problem 7
For nonnegative integers and with , let . Let denote the sum of all , where and are nonnegative integers with . Find the remainder when is divided by .
Let , and note that . The problem thus asks for the sum over all such that . Consider an array of 18 dots, with 3 columns of 6 dots each. The desired expression counts the total number of ways to select 6 dots by considering each column separately. However, this must be equal to . Therefore, the answer is .
Treating as , this problem asks for But can be computed through the following combinatorial argument. Choosing elements from a set of size is the same as splitting the set into two sets of size and choosing elements from one, from the other where . The number of ways to perform such a procedure is simply . Therefore, the requested sum is As such, our answer is .
Solution 3 (Major Bash)
Case 1: .
Subcase 1: Subcase 2: Subcase 3:
By just switching and in all of the above cases, we will get all of the cases such that is true. Therefore, this case is also
We begin as in solution 1 to rewrite the sum as over all such that . Consider the polynomial . We can see the sum we wish to compute is just the coefficient of the term. However . Therefore, the coefficient of the term is just so the answer is .
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