Chapter 2: Counting and Probability
Section 2.1-2.3
The Multiplication Principle and Permutations
Example 1 Consider the experiment of flipping a coin and then rolling a six-sided die.
How many possible outcomes are there?
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 tasks can be completed.
Example 2 Sonic has 25 different flavors of Milk Shake that each come in 3 different
sizes. If an order consists of first choosing which flavor of milk shake and then the size
of the shake, how many different orders are possible?
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 how the
first task was completed, 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.
Example 3 Texas license plates use a combination of three letters of the alphabet followed
by three digits from 1-9. How many different license plates can be issued using this
configuration?
Note: If there are restrictions on the tasks, then extra care is required when determining
the number of ways a task can be completed.
Example 4 Consider our previous example. How many different license plates can be
issued using this configuration if
a) no letter can be repeated?
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b) no digits can be repeated?
c) no letters or digits can be repeated?
Factorials: For any natural number n,
Note: 0! = 1
Calculating Factorials Using Your Calculator:
(1) n
(2) Press MATH
(3) Scroll to PRB
(4) Select ! (Option 4)
(5) Press ENTER
Example 5 Calculate 6!
A
of all the elements.
of a set of elements is an ordered arrangement
Example 6 Find the number of ways to arrange the elements in the set S = {a, b, c}.
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The number of permutations of n distinct objects is given by
Example 7 Daniel and Amber go to the movies with 5 of their friends and they all sit
together in the same row.
a) How many different ways can they arrange themselves?
b) How many different ways can they arrange themselves if Daniel and Amber sit next
to each other?
The number of permutations of r distinct objects taken from a set of size n is
given by
Calculating Permutations Using Your Calculator:
(1) Press the total number of elements in your set, n.
(2) Press MATH
(3) Scroll to PRB
(4) Select nP r (Option 2)
(5) Press the number of objects to be permuted, r.
(6) Press ENTER
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Example 8 The Texas Rangers have 14 batters on their roster. How many nine-person
batting orders can be formed?
Example 9 The Aggie Football team has 11 offensive starters. Five are linemen and six
are skill players. If a picture is taken with 3 linemen in the back row and 2 skill players
in the front row, how many different arrangements can there be?
Combinations
Example 10 Consider the set S = {a, b, c}. How many subsets with two elements are
there?
of r distinct objects taken from a set of
A
size n is a selection of r of the objects without concern for order.
The number of combinations of r distinct objects taken from a set of size n,
denoted by C(n, r), is given by
Calculating Combinations Using Your Calculator:
(1) Press the total number of objects available, n.
(2) Press MATH
(3) Scroll to PRB
(4) Select nCr (Option 3)
(5) Press the number of objects being selected, r.
(6) Press ENTER
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Example 11 A bag contains 5 red marbles, 3 green marbles, and 6 blue marbles.
a) In how many ways can a sample of 5 marbles be selected from the bag?
b) In how many ways can the sample consist of exactly 3 blue marbles?
c) In how many ways can the sample consist of at most 1 red marble?
d) In how many ways can the sample consist of at least 2 red marbles?
e) In how many ways can the sample contain only blue marbles?
f) In how many ways can the sample contain all of the green marbles?
g) In how many ways can the sample consist of exactly 2 red marbles or exactly 3 blue
marbles?
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Example 12 A student organization with 4 males and 6 females needs to elect a president,
vice president, treasurer, and secretary.
a) How many ways can this be done?
b) How may ways can it be done if there must be two males and two females selected?
Probability Applications of Counting Principles
Recall: The probability of an event E in some sample space S is given by
Example 13 A bag contains 5 red marbles, 3 green marbles, and 6 blue marbles. A
sample of 5 marbles is selected from the bag.
a) What is the probability that the sample consists of exactly 2 red marbles and 3 blue
marbles?
b) What is the probability that the sample consists of at most 1 red marble?
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c) What is the probability that the sample consists of at least 2 red marbles?
d) What is the probability that the sample contains only blue marbles?
e) What is the probability that the sample contains all of the green marbles?
f) What is the probability that the sample consists of exactly 2 red marbles or exactly 3
blue marbles?
Example 14 In a game of poker, a player is dealt 5 cards. A flush is a hand that consists
of a single suit. What is the probability that a player has a flush?
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Distinguishable Permutations: A set of size n is divided into k groups of sizes n1 , n2 ,
... , nk with all elements in each group being identical. Then the number of distinguishable
permutations is
where n1 + n2 + · · · + nk = n.
Example 15 How many distinguishable ways can the letters in the word TENNESSEE
be rearranged?
Example 16 Suppose you have 3 green marbles, 5 purple marbles, and 6 blue marbles.
How many distinguishable ways can you arrange these marbles in a row?
Example 17 There are 100 students in this class. What is the probability that at least 2
of them will have the same birthday?
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