2009 AIME I Problems/Problem 9

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We are given the 7 digits of the values, so let us find the ways we can distrubute these digits among the prizes. We are supposed to find the ways to group the digits such that each price has at least one digit, and no prices have 5 or more digits. We can express 7 this way as :

4, 2, 1 3, 3, 1 3, 2, 2

Let us begin with the first case. The possibilities are like so:

  • When we construct the table for the 1 and 2 digit numbers, we just permute the other 4 to find the choices for the 4 digit number

1 digit number 2 digit number 4 digit number

1 11 4 choose 1, or 4 numbers 1 13 4 choose 2, or 6 numbers 1 31 4 choose 2, or 6 numbers 1 33 4 choose 3, or 4 numbers 3 11 4 choose 2, or 6 numbers 3 13 4 choose 3, or 4 numbers 3 31 4 choose 3, or 4 numbers 3 33 4 choose 4, or 1 number


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