Mock AIME 3 Pre 2005 Problems/Problem 13
Let denote the value of the sum
Determine the remainder obtained when is divided by .
Let . Let . Then note that , so taking the derivative and multiplying by gives . Taking the derivative and multiplying by again gives . Now note that . Then we get , so , so .
Let the wanted sum be . We will simplify the expression into: . A counting argument will be provided to compute this.
Consider people, who will be separated into group , group , and group . Furthermore, one person in group will be cool, and one person will be smart. (They may be the same people).
Consider only putting people into group . There are ways this can be done. For the remaining people, there are one of two groups they can be in, namely group and group . This means that there are ways this can be done. There are ways to determine who is cool, and ways to determine who is smart. This is . As ranges from to , we will get all such scenarios. This means that the number of ways that this can be done is also .
Another way to count this is two split it up into two cases.
Case 1: person is both cool and smart. There are ways to choose this person. The remaining people have a choice of one of groups, making ways.
Case 2: person is cool, and another person in smart. There are ways to choose who is cool, and ways to choose who is smart. The remaining people have a choice of one of groups, making .
Thus, we have:
Thus, the answer is
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