2009 AIME I Problems/Problem 9

Problem

A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ 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 $1, 1, 1, 1, 3, 3, 3$ . Find the total number of possible guesses for all three prizes consistent with the hint.

Solution

[Clarification: You are supposed to find the number of all possible tuples of prices, $(A, B, C)$ , 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 $A=113, B=13, C=31$ , then the string is

$1131331.$

Since the strings have seven digits and three threes, there are $\binom{7}{3}=35$ 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

$x+y+z=7, x,y,z>0.$

This gives us

$\binom{6}{2}=15$

ways by stars and bars. But we have counted the one with 5 digit numbers; that is, $(5,1,1),(1,1,5),(1,5,1)$ .

Thus, each arrangement has $\binom{6}{2}-3=12$ ways per arrangement, and there are $12\times35=\boxed{420}$ ways.

Video Solution

https://youtu.be/VhyLeQufKr8

2009 AIME I (Problems • Answer Key • Resources)
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2009 AIME I Problems/Problem 9

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Video Solution

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