Goals
Notes
We extend the list of counting principles. Recall:
The Fundamental Counting Principle
If two or more events can occur in sequence, the total number of
ways they occur together is the product (multiplication) of the
number of ways each event occurs.
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Permutations
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Combinations
Arrangement matters for permutations!
Arrangement does not matter for combinations!
Permutations
Notes
An important application of the counting principle is the number of
ways to arrange some number of objects, n.
Permutations
The number of different ordered ways to arrange n distinct objects is
n! = n × (n − 1) × (n − 2) × (n − 3) × . . . × 3 × 2 × 1.
Here are several values of n!: 1! = 1, 2! = 2 · 1, 3! = 3 · 2 · 1, and
4! = 4 · 3 · 2 · 1. (0! = 1 is special.)
Ex: Sudoku The number of ways to arrange the digits 1 to 9 in a
single 3-by-3 square is
9! = 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 362, 880
Permutations
Ex: World cup FIFA.com lists 32 qualified teams from around the
world to participate in the 2014 World Cup. The total number of
different standings of the 32 distinct teams possible is the number
32!. This is about 2.6 × 1035 :
32! ≈ 26, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000
How many possible ways can be made for final matchup of two
teams?
How many possible ways can be made of the semi-finals of four
teams?
What pattern can you find in these numbers?
(Hint: Can you write the final number using only factorials?)
Notes
Permutations
Notes
Permutation of n objects taken r at a time
The number of permutations of n distinct objects taken r at a time is
n Pr
=
n!
, where r ≤ n
(n − r )!
Ex: Notation Perform the indicated calculation: 9 P5
Ex: World Cup How many possible ways can be made for final
matchup of two teams?
This is
32 P2
=
32 · 31 · 30!
32!
=
= 32 · 31 = 992
30!
30!
Ex: Secret codes Find the number of ways of forming four-digit
codes in which no digit (0 to 9) is repeated.
Permutations
Notes
Permutation of n objects ”picked” r at a time are used in:
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Final standings in a competition or race.
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Selecting positions (president, etc) from a board of members.
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Selecting a schedule of classes.
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Civil design.
Ex: Design Imagine planning a new shopping district of 12 stores. 6
are restaurants, 4 are clothing stores, and 2 are bookstores. In how
many distinguishable ways can the stores be arranged?
Distinguishable Permutations
The number of distinguishable permutations of n objects, with n1 are
of first type, n2 are of another type, etc, is
n!
where n1 + n2 + n3 + · · · + nk = n
n1 ! · n2 ! · n3 ! · · · nk !
Combinations
Ex: Design Imagine again planning a new shopping distict of 12 stores.
You can only advertise four of the stores on a billboard down the road.
In this selection, order does not matter. In how many combinations of four
stores can be ”chosen” from the twelve stores?
A permutation of 12 objects in which the 4 selected and (12 − 4) = 8 are
not selected is
12 C4
=
12!
12 · 11 · 10 · 9 · 8!
12 · 11 · 10 · 9
=
=
= 495
4! · 8!
4! · 8!
4!
Combinations of n objects taken r at a time
A combination is a selection of r objects from n objects and disregards
order. The number is
n!
n Cr =
(n − r )! · r !
Notes
More Examples
Notes
In these problems, ask if order matters or if order does not matter.
Then either use permutations or combinations.
Ex: A manager needs to select a three-person team from 20 people.
In how many ways can the manager form this team?
Ex: A manager needs to nominate a president, secretary, and
webmaster from 20 people. In how many ways can the manager
nominate a group?
Ex: You need to arrange 6 letters consist of one L, two E’s, two T’s,
and one R. The letters are randomly ordered. What is the probability
that the arrangement spell the word letter?
More Examples
Ex: Notation Perform the indicated calculation:
Ex: Notation Perform the indicated calculation:
Notes
6 P2
11 P3
12 C4
52 C5
Ex: Probability A sample of 400 corn kernels has three kernels with
high levels of a toxin. Four kernels are selected randomly. What is the
probability that exactly one kernel contains a high level of toxin?
Ex: Probability A jury consists of five men and seven women. Three
are selected at random for an interview. Find the probability that all
three are men.
Assignment
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Homework §3.4 # 8, 10, 12, 14, 30, 38, 48, 56
Study page 185 for the exam. Try the exercises on page 181 to 184
for extra review.
Notes