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 <dollar/> to <dollar/> 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.

Solution

Since we have 3 numbers, consider how many ways we can put this 3 number in a string of 7 digits by putting A,B,C together

For example:

Then the string is

Since the strings have 7 digits and 3 three's. There are

of such string

In other to obtain all combination of A,B,C. We partition all the possible strings into 3 groups

Let look at the example.

We have to partition it into 3 groups with each group having at least 1 digit

We have to find solution where

This gives us:

(balls and urns)

But we have counted the one with 5 digit numbers. That is

Thus, each arrangement has ways per arrangement

Thus, there are ways.