8 - Counting

Summary

Introduction

This chapter considers problems that involve permutations, combinations, partitioning, and other counting-oriented problems. Some AMC problems involve nothing more than the application of these ideas, others use these counting techniques as a first step when solving a more complicated problem.

Permutations

DEFINITION 1 A permutation of a collection of distinguishable objects is an arrangement of the objects in some specific order.

For example, acbd and dabc are both permutations of the letters a, b, c, and d.What generally interests us is the number of different permutations that are possible from a given collection. In this case there are 24 different permutations of these 4 letters. This is because any one of the 4 letters could be first. Then there are 3 possible choices remaining for the second letter, 2 possible choices remaining for the third, and, of course, only one choice remaining for the last. This is a special case of the following general result.

RESULT 1 The number distinct permutations of N distinguishable objects is

N · (N − 1) · (N − 2) · · · 2 · 1 = N!

When some of the objects are not distinguishable, the number of distinct permutations is reduced. Suppose that we want the number of distinct permutations of the five letters a, a, b, c, and d. If the letters were distinguishable the number would be 5!. However, we cannot distinguish between the two a's so we must reduce the number by a factor of 2, since the first time an a appeared it could be any one of the 2 possibilities. In a similar manner, if the set of letters was a, a, a, b, and c. We would need to reduce the number by a larger factor 3! since the first time an a appeared in a permutation, it could be any one of the three possibilities, the second a appearing could be any of the 2 remaining a's, and there would be only one possibility remaining for the third appearing a. This logic leads to a general permutation result for collections of objects not all of which can be distinguished.