Permutations

1
Section 12.8

• Permutations

2
Definition of a Permutation

• A permutation is an arrangement of objects,
• where the order of arrangement matters.
• Examples of Permutations Lining up for a
• photo Exacta in a Horse Race
• Example that is Not a Permutation
• Lottery

3
With Replacement Repetition Permitted
A user ID for an internet ticket-buying service
is required to be 4 letters followed by 2
numbers. How many different user IDs are
possible, if repeated letters and numbers are
OK? The scheme is __ __ __ __ __ __
L L L L N N

4
Counting Principle and Permutations
The number of choices for each position are
26 26 26 26 10 10
L L L L N
N The number of permutations is therefore
26x26x26x26x10x10 45,697,600

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Without Replacement No Repetition
How does the number of permutations change if
repetition is not permitted?
26 25 24 23 10 9
L L L L N N The number of
permutations will be less than before 26x25x24x2
3x10x9 32,292,000

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Permutations Without Replacement

• Experiment There are 4 names in a hat. In how
  many different ways can we draw the 4 names out
  of the hat, without replacement?
• The number of permutations is
• (4)(3)(2)(1) 24

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Factorial

• (4)(3)(2)(1) 24 is called 4 factorial and
  is
• written as 4!
• Factorial is defined as
• n! n(n1)(n2)?1
• 3! 6 5! 120 0! is
  defined to be 1

8
Arrange 5 from 9
Suppose you have 9 paintings by an artist.
Your gallery is going to hang 5 of them in an
exhibit. How many different ways can 5 paintings
be arranged out of 9 ? Use the Counting
Principle 9 x 8 x 7 x 6 x 5 15,120
permutations

9
Permutation Formula
9 ? 8 ? 7 ? 6 ? 5
Permutation formula gives the number of
different ways of selecting r objects out of n
total

10
Permutation Formula
A basketball team roster has 14 players.
How many different photographs showing 5 players
are possible? Use the permutation formula
or the permutation key on your calculator to
calculate 14 P5 14!
240,240 different arrangements.
9!