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CHAPTER 8 COMBINATIONS AND PERMUTATIONS LEARNING
OBJECTIVES Upon completion of this chapter, you should be able to do
the following: 1. Define combinations and permutations. 2. Apply the concept of combinations to problem solving. 3. Apply the concept of principle of choice to problem
solving. 4. Apply the concept of permutations to problem solving. INTRODUCTION This chapter deals with concepts required for the study of
probability and statistics. Statistics is a branch of science that is an
outgrowth of the theory of probability. Combinations and per mutations are used in both statistics and probability; and
they, in turn, involve operations with factorial notation. Therefore,
combinations, permutations, and factorial notation are discussed in this
chapter. DEFINITIONS A combination is defined as a possible selection
of a certain number of objects taken from a group without regard to order. For
instance, suppose we were to choose two letters from a group of three letters. If the group of three letters were A, B,
and C, we could choose the letters in combinations of two as follows: AB, AC, BC The order in which we wrote the letters is of no concern;
that is, AB could be written BA, but we would still have only
one combination of the letters A and B. A permutation is defined as a possible selection of a
certain number of objects taken from a group with regard to order. The
permutations of two letters from the group of three letters would be as
follows: AB, AC, BC, BA, CA, CB The symbol used to indicate the foregoing combination will
be ,C, meaning a group of three objects taken two at a time. For the previous
permutation we will use
, meaning a group of three objects taken two at a
time and ordered. You will need an understanding of factorial notation before
we begin a detailed discussion of combinations and permutations. We define the
product of the integers n through 1 as n factorial and use the symbol n! to
denote this; that is,
EXAMPLE: Find the value of 5! SOLUTION:
EXAMPLE: Find the value of
SOLUTION:
and
Then
and by simplification
The previous example could have been solved by writing
Notice that we wrote
and combine the factors
as 3! so that
EXAMPLE: Find the value of
SOLUTION:
and
Then
Notice that 4! was factored from the expression 6!  4! Theorem. If n and r are positive integers,
with n greater than r_{2} then
This theorem allows us to simplify an expression as
follows:
Another example is
EXAMPLE: Simplify
SOLUTION:
then
PRACTICE PROBLEMS: Find the value of problems 1 through 4 and simplify
problems 5 and 6.
ANSWERS:
