Permutations

1
Permutations
2
Basics
5!
Does not mean FIVE!
5! is read five factorial.
5!
5 4 3 2 1
7!
7 6 5 4 3 2 1
5! 120
7! 5040
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When a group of objects or things are arranged
in a certain order, the arrangement is called a
permutation.
The arrangement of objects in a line is called a
linear permutation.
4
Example 1.
A theater owner has 11 films to show on 8
different screens. How many different
arrange- ments are there for the 8 screens to
show the 11 movies.
5
Example 1.
8 screens and 11 movies.
We must make 8 choices.
1
2
3
4
Choice
11
10
9
8

Choice
5
6
7
8
7
6
5
4

6
Example 1.
1
2
3
4
Choice
11
10
9
8

Choice
5
6
7
8
7
6
5
4

Therefore by the FCP there are 11 10 9 8 7
6 5 4
7
Example 1.
Therefore by the FCP there are 11 10 9 8 7
6 5 4 arrangements.
The number of way to arrange 11 objects taken 8
at a time is written as P(11,8).
8
P(n,r) is read as the permutation of n objects
taken r at a time.
Definition of P(n,r)
P(n,r)
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Example 2.
Five teens find seven empty seats at a theater.
How many different seating arrangements are
there?
P(7,5)

10
Example 2.
P(7,5)


7 6 5 4 3
2520
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Example 3.
A music store manager want to arrange 5 rock
CDs 4 rap CDs and 4 jazz CDs on a shelf.
How many ways can they be arranged if they are
ordered according to type?
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Example 3.
The 5 rock CDs arrange into P(5,5) or 5! ways.
The 4 rap CDs arrange into P(4,4) or 4! ways.
The 4 jazz CDs arrange into P(4,4) or 4! ways.
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Example 3.
There are 3 types of CDs that arrange into
P(3,3) or 3! ways.
The total ways the CDs can be arranged is the
product.
P(5,5)P(4,4)P(4,4)P(3,3)
414,720
5!4!4!3!
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Permutations with repetitions.
The number of permutations of n objects of which
p are alike and q are alike is
The rule can be extended to any number of objects
that repeat.
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Example 4.
How many ways can the letters of the word
perpendicular be arranged?
There are 2 ps, 2 es, and 2 rs.
Therefore
arrangements
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Example 4.
How many ways can the letters of the word
perpendicular be arranged?
778,377,600
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Circular Permutations.
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Example 5.
A disc jockey is loading a circular tray with
six compact discs. How many different ways can
these be arranged?
(n-1)! (6-1)! 5! 120
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Circular Permutations (cont).
If n distinct objects are arranged in a circle
and there is a fixed point on the circle then
there are n! permutations of the objects around
the circle.
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Example 6.
If five different homes are being built around a
cul-de-sac. How many different arrangements of
the homes are possible?
The cul-de-sac is circular but the entrance road
is a fixed point
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Example 6.
If five different homes are being built around a
cul-de-sac.
The cul-de-sac is circular but the entrance road
is a fixed point
There are n! 5! 120 arrangements possible.
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We now have several formulas to remember.
Linear permutations, circular permutations,
circular permutations with a fixed point, and
permutations with repetitions.
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One last thing we need to know.
Reflection happens when a circular permutation
can be flipped over or when a linear
permutation is viewed from opposite sides. This
causes the number of permutations to be half as
many.