1984 AIME Problems/Problem 11
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 notice that there is no difference between the maple trees and the oak trees; we have only two types, birth trees and "non-birch" trees.
The five birch trees must 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.
There are 
total ways to arrange the twelve trees, so the probability is 
.
The answer is 
.
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 | ||
