Difference between revisions of "1990 AIME Problems/Problem 8"
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2) The marksman must then break the lowest remaining target in the chosen column. | 2) The marksman must then break the lowest remaining target in the chosen column. | ||
| − | If the rules are followed, in how many different | + | If the rules are followed, in how many different [[order]]s can the eight targets be broken? |
== Solution == | == Solution == | ||
| − | {{ | + | Suppose that the columns are labeled <math>A</math>, <math>B</math>, and <math>C</math>. The question is asking for the number of ways to shoot at the bottom target of the columns, or the number of ways to arrange 3 <math>A</math>s, 3 <math>B</math>s, and 2 <math>C</math>s in a string of 8 letters. |
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| + | Of the 8 letters, 3 of them are <math>A</math>s, making <math>{8\choose3}</math> possibilities. Of the remaining 5, 3 are <math>B</math>s, making <math>{5\choose3}</math> possibilities. The positions of the 2 <math>C</math>s are then fixed. Thus, there are <math>{8\choose3} \cdot {5\choose3} = 560</math> ways of shooting all of the targets. | ||
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| + | Alternatively, the number of ways to arrange 3 <math>A</math>s, 3 <math>B</math>s, and 2 <math>C</math>s in a string of 8 letters is equal to <math>\frac{8!}{3! \cdot 3! \cdot 2!} = 560</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=1990|num-b=7|num-a=9}} | {{AIME box|year=1990|num-b=7|num-a=9}} | ||
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| + | [[Category:Intermediate Combinatorics Problems]] | ||
Revision as of 10:01, 3 March 2007
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
Suppose that the columns are labeled 
, 
, and 
. The question is asking for the number of ways to shoot at the bottom target of the columns, or the number of ways to arrange 3 s, 3 s, and 2 s in a string of 8 letters.
Of the 8 letters, 3 of them are s, making 
possibilities. Of the remaining 5, 3 are s, making 
possibilities. The positions of the 2 s are then fixed. Thus, there are 
ways of shooting all of the targets.
Alternatively, the number of ways to arrange 3 s, 3 s, and 2 s in a string of 8 letters is equal to 
.
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 | ||
