Section 11.2: Permutations
Permutations
Permutation is an ordered arrangement of items that occurs when:
• No item is used more than once.
• The order of arrangement makes a difference.
Example
Counting Permutations
You need to arrange seven of your favorite books along a small shelf. How many different
ways can you arrange the books, assuming that the order of the books makes a
difference to you?
Solution:
You can choose any one of the seven books for the first position on the shelf. This leave six
choices for the second position. After the first two positions are filled, there are five books
to choose and so on.
7  6  5  4  3  2  1 = 5040
Factorial Notation n!
The product 7∙6∙5∙4∙3∙2∙1 is called 7 factorial and is written 7!
Definition:
If n is a positive integer, the notation n! (read “n factorial”) is the product
of all positive integers from n down through 1.
n! = n(n  1)(n  2)∙∙∙(3)(2)(1)
0! (zero factorial), by definition, is 1.
0! = 1
Find Factorial n! on Calculators
Press
and press 3 for x!.
Press PRB and use
to select !
Press MATH and use
Use Enter and use
to go to PRB.
to go to 4: !
Example Using Factorial Notation
Evaluate with using your calculator:
1. 9! = 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 362880. Use your calculator function
to get this answer directly.
2. 5!
= 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120. Use your calculator function to get this answer
directly.
Example Using Factorial Notation
Evaluate without using your calculator: 𝑎.
8!
5!
𝑏.
26!
21!
8! 8 ⋅ 7 ⋅ 6 ⋅ 5!
𝑎. =
= 8 ⋅ 7 ⋅ 6 = 336
5!
5!
Cancel common factor 5! on Denominator and Nominator.
26! 26 ⋅ 25 ⋅ 24 ⋅ 23 ⋅ 22 ⋅ 21!
𝑏.
=
= 26 ⋅ 25 ⋅ 24 ⋅ 23 ⋅ 22
21!
21!
= 7,893,600
Cancel common factor 21! on Denominator and Nominator.
A Formula for Permutations
The number of possible permutations of r items are taken from n items is:
n!
n Pr 
(n  r )!
Press 2nd and press nPr
Press PRB and use
to select nPr
Press MATH and use
Use Enter and use
to go to PRB.
to go to 2: nPr
Example
Using Permutations function in your calculator
Evaluate using your calculator: 𝑎. 8𝑃5
𝑎. 8𝑃5 = 672
b.
26𝑃3
= 15600
𝑏. 26𝑃3
𝑛𝑃𝑟
Example Using the Formula for Permutations
You and 19 of your friends have decided to form a business. The group needs
to choose three officers– a CEO, an operating manager, and a treasurer. In
how many ways can those offices be filled?
Solution:
Your group is choosing r = 3 officers from a group of n = 20 people. The order
matters because each officer has different responsibilities:
20!
20! 20 ⋅ 19 ⋅ 18 ⋅ 17!
=
=
= 20 ⋅ 19 ⋅ 18 = 6840
𝑛𝑃𝑟 = 20𝑃3 =
(20 − 3)! 17!
17!
You can also use the calculator function to get 6840 after use nPr function.
Example
Using the Formula for Permutations
How many arrangements can be made using four of the letters of
the word COMBINES if no letter is to be used more than​ once?
Solution:
There are 8 letters and use 4 from these letters to make a new word.
Use nPr to get it.
P4= 1680
8
There are 1680 arrangement.
A Formula for Permutations of Duplicate Items
Permutations of Duplicate Items:
The number of permutations of n items, where p items are identical, q items
are identical, r items are identical, and so on, is given by:
n!
p !q !r !...
Example
Using the Formula for Permutations of Duplicate Items
In how many distinct ways can the letters of the word MISSISSIPPI be arranged?
Solution:
The word contains 11 letters (n = 11) where four Is are identical (p = 4), four Ss are identical
(q = 4) and 2 Ps are identical (r = 2). The number of distinct permutations are:
n!
11!
1110  9  8  7  6  5  4!


 34,650
p!q!r ! 4!4!2!
4! 4  3  2 1 2 1
Example
Using the Formula for Permutations of Duplicate Items
A signal can be formed by running different colored flags up a​ pole, one
above the other. Find the number of different signals consisting of 9
flags that can be made using 4 white​ flags, 3 red​ flags, and 2 blue flags.
Solution: The number of distinct permutations are:
𝑛!
9!
9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4!
=
=
= 1260
𝑝! 𝑞! 𝑟! 4! 3! 2! 4! ⋅ 3 ⋅ 2 ⋅ 1 ⋅ 2 ⋅ 1
1260 different signals