6.7 Day 1 Permutations 2011
January 28, 2011
6.7 Day 1 ‐ Permutations
Objectives
Use permutations to solve counting problems.
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6.7 Day 1 Permutations 2011
January 28, 2011
Warm‐up:
1. Given the word HEAR, how many
possible ways are there to arrange the
letters?
2. Now take the word HEART. How
many possible ways are there to arrange the
letters?
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6.7 Day 1 Permutations 2011
January 28, 2011
You were given the word HEAR.
What happened when you changed the word to HEART.
What was your strategy?
What was your strategy?
How many arrangements did you come up with?
How many arrangements did you come up with?
Did you get 24?
Did you get 120?
Is there a pattern??? Is there a trick?? What is it?
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6.7 Day 1 Permutations 2011
H
January 28, 2011
E
A
R
There are 4 possible choices for the first letter: H, E, A, and R.
After one of these letters is put in the first space, there are only 3 choices left.
After you have filled spaces 1 and 2, there are only 2 letter choices left.
After you have filled spaces 1, 2, and 3, there is only 1 letter left.
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6.7 Day 1 Permutations 2011
January 28, 2011
Multiplication Counting Principle: The number of possible
outcomes for an event is found by multiplying the number
of choices at each stage of the event.
4
X
3
X
2
4 possible choices for the first space
1 = 24
H EA R
X
3 possible choices for the second space
2 possible choices for the third space
1 possible choice for the last space
So there are 24 possible
arrangements
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6.7 Day 1 Permutations 2011
January 28, 2011
Then you added a T. This gave the word HEART. Using the method that we just did, how many different 5­letter arrangements can be made from the letters of HEART?
5 x 4 x 3 x 2 x 1 = 120
H E A RT
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6.7 Day 1 Permutations 2011
January 28, 2011
How many different 2­letter arrangements can be made from the letters in the word GRATES?
G R A TE S
6 x 5 = 30 7
6.7 Day 1 Permutations 2011
January 28, 2011
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6.7 Day 1 Permutations 2011
January 28, 2011
The symbol ! is called factorial and is used in mathematics in a specific way.
You read the symbol 6! as "six factorial". This is what it means:
6! = 6 x 5 x 4 x 3 x 2 x 1
Start with the number given to you
Mulitply by all the smaller numbers
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6.7 Day 1 Permutations 2011
January 28, 2011
Permutations
The arrangement of any number of items in a definite order is called a permutation.
The symbol for the number of different arragements when n items are arranged r at a time is nPr
In the example with HEART, 5 items (letters) are arranged 5 at a time. We used the muliplication counting principle to find the number of possible arrangements.
=
P
5 5
5 items
5 x 4 x 3 x 2 x 1 = 120
arranged 5 at a time
There are 5 letters, and we are arranging 5 of them
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6.7 Day 1 Permutations 2011
January 28, 2011
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6.7 Day 1 Permutations 2011
January 28, 2011
We did an example with the word GRATES. We had 6 items (letters) arranged 2 at a time.
P
= 6 x 5 = 30 6 2
There are 6 letters, and we are arranging 2 of them
You can also write this using factorials.
6 x 5 x 4 x 3 x 2 x 1 P
6 x 5 =
= 4 x 3 x 2 x 1 =
6 2
6!
6!
=
4! (6­2)!
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6.7 Day 1 Permutations 2011
January 28, 2011
In general,
P
n r
=
n!
(n­r)!
An example,
P = 10 x 9 x 8
10 3
10!
10!
=
=
(10­3)!
7!
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6.7 Day 1 Permutations 2011
January 28, 2011
How many different ways can you arrange a license plate if the first 3 positions are letters and the next 4 positions are numbers 0­9?
Without repeats,
26
Letter 1
25
Letter 2
24
10
9
8
7
Letter 3
#1
#2
#3
#4
10
10
#3
#4
26 x 25 x 24 x 10 x 9 x 8 x 7 = 78,624,000
With repeats,
26
Letter 1
26
Letter 2
26
10
Letter 3
#1
10
#2
26 x 26 x 26 x 10 x 10 x 10 x 10 = 175,760,000
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6.7 Day 1 Permutations 2011
January 28, 2011
Homework
page 348 (1 ­ 20)
&
Counting Calamari wkst
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