MATH 166 Spring 2016
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Chapter 2 Counting and Probability
2.1 The Multiplication Principle and Permutations
Multiplication Principle: Suppose there is an operation that consists of making a sequence of
two tasks with n1 choices for the first task and, no matter how the first task was completed, there
are always n2 possible choices for the second task. Then there are n1 · n2 possible ways in which the
sequence of two task can be completed.
General Multiplication Principle: Suppose there is an operation that consists of making a sequence of k tasks with n1 possible choices for the first task and, no matter what choice was made for
the first task, there are always n2 possible choices for the second task, and, no matter what choices
were made for the first two tasks, there are always n3 possible choices for the third task, and so on.
Then there are n1 · n2 · · · nk possible ways in which the sequence of k tasks can be made.
Examples:
1. A contractor has eight styles of homes, each with four styles of garages and six styles of decks.
How many different possible homes are there?
2. How many five-digit telephone numbers are possible if the first digit must be nonzero?
3. How many international direct-dialing numbers for calls within the United States and Canada
are possible if each number consists of a 1 plus a three-digit area code (the first digit of which
must be nonzero) and a number of the type described in problem 2?
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Factorials: For any natural number n
n! = n(n − 1)(n − 2) · · · 3 · 2 · 1
0! = 1
Definition: A permutation of a set of elements is an ordered arrangement of all the elements.
Number of Permutations of n Objects: The number of permutations of n distinct objects is
given by
P (n, n) = n!
Number of Permutations of r Objects Taken from a Set of Size n: The number of permutations of r distinct objects taken from a set of size n is given by
P (n, r) =
n!
= n(n − 1)(n − 2) · · · (n − r + 1)
(n − r)!
Technology Corner:
• To compute n!: enter n, press MATH , arrow over to the PRB menu, and choose 4: !
• To compute P (n, r): enter n, press MATH , arrow over to the PRB menu, choose 2: nPr, and
enter r
Examples:
1. (p. 87) In how many ways can a football team of 11 players arrange themselves to trot onto
the football field one at a time?
2. A certain state uses license plates with three letters followed by three digits.
(a) How many license plates are possible?
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(b) If no repetition of letters or digits is permitted, how many different license plates are
possible?
(c) How many license plates have no repetition of letters or digits and begin with a vowel?
3. An exam consists of three true/false questions followed by four multiple choice questions each
with four answers.
(a) How many ways can a student answer the exam if they answer all of the questions?
(b) How many ways can a student answer the exam if they can leave true/false questions
blank?
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4. A company car that has a seating capacity of six is to be used by six employees who have formed
a car pool. If only three of these employees can drive, how many possible seating arrangements
are there for the group?
5. Five men and eight women are to line up for a picture with the nine men in the middle. How
many ways can this be done? (Assume there are two women on each side of the group of men)
6. At a college library exhibition of faculty publications, 4 mathematics books, 4 social science
books, and 3 biology books will be displayed on a shelf. (Assume that none of the books are
alike.)
(a) In how many ways can the 11 books be arranged on the shelf?
(b) In how many ways can the 11 books be arranged on the shelf if books on the same subject
matter are placed together?
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7. In how many ways can 2 married couples attending a concert be seated in a row of 4 seats if
(a) There are no restrictions?
(b) Each married couple is seated together?
(c) The members of each sex are seated together?
8. Find the number of distinguishable arrangements of each of the following ”words.”
(a) acdbes
(b) baaabe
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(c) aaabba
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