# 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 $\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $\displaystyle m+n$.

## Solution

First off, notice that there are ${7\choose4} = 35$ 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 ${8\choose5} = 56$ different ways to arrange this. Thus in total, there are $35 \cdot 56 = 1960$ such arrangements.

There are a total of $\frac{(3 + 4 + 5)!}{3! 4! 5!} = 27720$ ways of arranging the trees, so the requested probability is $\frac{1960}{27720} = \frac{7}{99}$, and $\displaystyle m+n = 106$.

## See also

 1984 AIME (Problems • Answer Key • Resources) Preceded byProblem 10 Followed byProblem 12 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All AIME Problems and Solutions
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