2009 AIME I Problems/Problem 9
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from to inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were . Find the total number of possible guesses for all three prizes consistent with the hint.
[Clarification: You are supposed to find the number of all possible tuples of prices, , that could have been on that day.]
Since we have three numbers, consider the number of ways we can put these three numbers together in a string of 7 digits. For example, if , then the string is
Since the strings have seven digits and three threes, there are arrangements of all such strings.
In order to obtain all combination of A,B,C, we partition all the possible strings into 3 groups.
Let's look at the example. We have to partition it into 3 groups with each group having at least 1 digit. In other words, we need to find the solution to
This gives us
ways by balls and urns. But we have counted the one with 5 digit numbers; that is, .
Thus, each arrangement has ways per arrangement, and there are ways.
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