Problem:
A gardener plants three maple trees, four oak trees 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:
The trees can be planted in orders. Let be the number of orders in which no two birch trees are adjacent to one another. The probability we need is . To find , we will count the number of patterns
where the 's denote nonbirch (i.e., maple and oak) trees, and slots through are to be occupied by birch trees, at most one in each slot. There are orders for the nonbirch trees, and for each ordering of them there are ways to place the birch trees. Thus, we find that and .
The problems on this page are the property of the MAA's American Mathematics Competitions