Difference between revisions of "1990 AIME Problems/Problem 8"
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From left to right, suppose that the columns are labeled <math>L,M,</math> and <math>R,</math> respectively. | From left to right, suppose that the columns are labeled <math>L,M,</math> and <math>R,</math> respectively. | ||
| − | Consider the string <math>LLLMMRRR</math> | + | Consider the string <math>LLLMMRRR.</math> Since the set of all letter arrangements is bijective to the set of all shooting orders, the answer is the number of letter arrangements, which is <math>\frac{8!}{3! \cdot 2! \cdot 3!} = \boxed{560}.</math> |
~Azjps (Solution) | ~Azjps (Solution) | ||
~MRENTHUSIASM (Revision) | ~MRENTHUSIASM (Revision) | ||
| + | |||
| + | == Remark == | ||
| + | We can count the letter arrangements of <math>LLLMMRRR</math> in two ways: | ||
| + | <ol style="margin-left: 1.5em;"> | ||
| + | <li></li><p> | ||
| + | <li></li><p> | ||
| + | </ol> | ||
| + | ~MRENTHUSIASM | ||
==Video Solution== | ==Video Solution== | ||
Revision as of 11:20, 19 June 2021
Problem
In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules:
1) The marksman first chooses a column from which a target is to be broken.
2) The marksman must then break the lowest remaining target in the chosen column.
If the rules are followed, in how many different orders can the eight targets be broken?
Solution
From left to right, suppose that the columns are labeled 
and 
respectively.
Consider the string 
Since the set of all letter arrangements is bijective to the set of all shooting orders, the answer is the number of letter arrangements, which is 
~Azjps (Solution)
~MRENTHUSIASM (Revision)
Remark
We can count the letter arrangements of 
in two ways:
~MRENTHUSIASM
Video Solution
https://www.youtube.com/watch?v=NGfMLCRUs3c&t=7s ~ MathEx
See also
| 1990 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
