Difference between revisions of "1984 AIME Problems/Problem 11"
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== Problem == | == Problem == | ||
| − | A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let <math>\frac m n</math> in lowest terms be the [[probability]] that no two birch trees are next to one another. Find <math> | + | A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let <math>\frac m n</math> in lowest terms be the [[probability]] that no two birch trees are next to one another. Find <math>m+n</math>. |
== Solution == | == Solution == | ||
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The five birch trees must now be placed amongst the seven previous trees. We can think of these trees as 7 dividers of 8 slots that the birch trees can go in, making <math>{8\choose5} = 56</math> different ways to arrange this. Thus in total, there are <math>35 \cdot 56 = 1960</math> such arrangements. | The five birch trees must now be placed amongst the seven previous trees. We can think of these trees as 7 dividers of 8 slots that the birch trees can go in, making <math>{8\choose5} = 56</math> different ways to arrange this. Thus in total, there are <math>35 \cdot 56 = 1960</math> such arrangements. | ||
| − | There are a total of <math>\frac{(3 + 4 + 5)!}{3! 4! 5!} = 27720</math> ways of arranging the trees, so the requested probability is <math>\frac{1960}{27720} = \frac{7}{99}</math>, and <math> | + | There are a total of <math>\frac{(3 + 4 + 5)!}{3! 4! 5!} = 27720</math> ways of arranging the trees, so the requested probability is <math>\frac{1960}{27720} = \frac{7}{99}</math>, and <math>m+n = 106</math>. |
== See also == | == See also == | ||
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* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
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| + | [[Category:Intermediate Combinatorics Problems]] | ||
Revision as of 08:56, 18 October 2007
Problem
A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let 
in lowest terms be the probability that no two birch trees are next to one another. Find 
.
Solution
First off, notice that there are 
ways of arranging just the maple and oak trees while ignoring the birch trees.
The five birch trees must now be placed amongst the seven previous trees. We can think of these trees as 7 dividers of 8 slots that the birch trees can go in, making 
different ways to arrange this. Thus in total, there are 
such arrangements.
There are a total of 
ways of arranging the trees, so the requested probability is 
, and 
.
See also
| 1984 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
