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Exercise
How many different lunches
can be made by choosing
one of four sandwiches, one
of three fruits, and one of
two desserts?
24
Exercise
How many different lunches
can be made by choosing
one of four sandwiches, one
of three fruits, one of two
desserts, and one of two
beverages?
48
Exercise
How many ways can four
separate roles be filled if
four people try out?
24
Exercise
How many ways can four
separate roles be filled if
seven people try out?
840
Exercise
How many ways can seven
separate roles be filled if
seven people try out?
5,040
Permutation
A permutation is a way of
arranging r out of n objects
(if r ≤ n).
Fundamental Principle of
Counting
If there are p ways that a first
choice can be made and q
ways that a second choice
can be made, then there are
p × q ways to make the first
choice followed by the
second choice.
n Factorial
The product of n natural
numbers from n down to one
is called n factorial. The
symbol for “factorial” is an
exclamation mark, and
n! = n(n – 1) … (1).
(0! = 1 by definition.)
Example 1
Evaluate 6!
6(5)(4)(3)(2)(1) = 720
Example
Evaluate 5!
120
Example
Evaluate 8!
40,320
Formula for the Permutation
of n Objects Taken n at a
Time—nPn
To find the number of
permutations of n distinct
objects taken n at a time, find
the product of the positive
integers n down through one:
nPn = n(n – 1)(n – 2) … (2)(1) = n!
Example 2
Find the number of
permutations of the letters
in the word saved.
Formula for the Permutation
of n Objects Taken r at a
Time—nPr
To find the number of
permutations of n distinct
objects taken r at a time, use
n!
the formula nPr = (n – r)! .
Example 3
Find the number of
permutations of the five
letters s, a, v, e, and d taken
three at a time.
5
x
4
x
3
x
2
x
1
5!
P
=
5 3 =
2x1
(5 – 3)!
= 5 x 4 x 3 = 60
Example 4
Find the number of
permutations of eight distinct
things taken three at a time.
8!
8!
=
8P3 =
(8 – 3)! 5!
8(7)(6)(5)(4)(3)(2)(1)
=
(5)(4)(3)(2)(1)
= (8)(7)(6) = 336
Example
Evaluate 5P3.
60
Example
Evaluate 9P4.
3,024
Example
Find the number of ways ten
books can be arranged on a
bookshelf.
10P10 =
10! = 3,628,800
Example
Find the number of possible
class schedules for a
student taking six different
classes.
6P6 =
6! = 720
Example
Find the number of different
ways of choosing 17
committee chairpersons
from the 51 senators in the
majority party.
27
P
=
51!
÷
34!
=
5.25
×
10
51 17
Example
In a race involving six
people, how many different
orders are possible for the
top three finishers?
6P3 =
6! ÷ 3! = 120
Example
How many ways are there to
select a jury foreman and
subforeman from among the
twelve jurors?
12P2 =
12! ÷ 10! = 132
Example
Find the number of ways of
arranging ten books on a
bookshelf if five are math
books and five are history
books and each category
must be grouped together.
5! x 5! x 2 = 28,800
Exercise
Write an expression in the
form nPr to represent the
number of permutations in
the following situations.
Exercise
Select first, second, and
third place out of 500
contestants.
500P3
Exercise
Elect a president, vice
president, secretary, and
treasurer from a class of
twenty-eight students.
28P4
Exercise
How many different twodigit whole numbers can
you make from the digits
2, 4, 6, and 8 if no digit
appears more than once in
each number?
4P2
Exercise
How many different threeletter arrangements are
there of the letters of the
alphabet?
26P3