Difference between revisions of "2009 AIME I Problems/Problem 9"
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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 | 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: <math>A=113, B=13, C= | + | For example: <math>A=113, B=13, C=31</math> |
Then the string is | Then the string is | ||
| − | <cmath> | + | <cmath>1131331</cmath> |
Since the strings have 7 digits and 3 three's. There are | Since the strings have 7 digits and 3 three's. There are | ||
Revision as of 12:14, 30 August 2009
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
![\[1131331\]](http://latex.artofproblemsolving.com/b/f/1/bf1fe2997c5a51f2f7232634051760abb2423917.png)
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
![\[x+y+z=7, 0<x,y,z<5\]](http://latex.artofproblemsolving.com/e/4/4/e44c4252cf16a3c626661cf9a88fa1e9cbd36f7f.png)
This gives us:
![\[_6C_2\]](http://latex.artofproblemsolving.com/b/b/e/bbeca8d4ef73c6e06c5233e56e80a8d7e2638268.png)
(balls and urns)
But we have counted the one with 5 digit numbers. That is 
Thus, each arrangement has ![\[(_6C_2)-3=12\]](http://latex.artofproblemsolving.com/c/c/8/cc8544ebab89906928daf06d1f95088e0a944749.png)
ways per arrangement
Thus, there are 
See also
| 2009 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
