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Permutations
• We use permutations when we are interested in the number of possible ways to order something and ORDER IS IMPORTANT!
• When order is not important, then it is a combination
n – total number of objects to choose from
r – number of times you choose an objectThus a permutation is an ordered arrangement of
“r” objects from a group of “m” objects.2
Permutation example
Suppose I have 8 different colors of gumballs. How many different ways can I give 3 children a gumball?
8 * 7 * 6 = 336
3
Permutation example
Does this look like we simply used the BCR? - BCR and Permutation are related. - Let’s find out why using the permutation formula:
!5
!8
)!38(
!838
P
8*7*6*5*4*3*2*1336
5*4*3*2*1
3366*7*8
4
Sampling Replacement
The outcome of a permutation depends on two things. Do we sample:
With replacement?or
Without replacement?
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Permutations with Replacement
These aren’t popular, but let’s see what one might look
like:
How many possible ways could I select from the letters
PURDUE if I sample with replacement?
_ _ _ _ _ _
6*6*6*6*6*6 = 66= 46656
6
Permutations without Replacement
How many different letter arrangements can be formed using the letters BOILERS?
Theorem: Special Permutations Rule
is the number of ways to order n distinct objects
_ _ _ _ _ _ _
So, there are 7! = 5040 ways to arrange the letters BOILERS.
!nPnn
7
Theorem
There are:
different permutations of m objects
of which m1, m2, … mk are alike
respectively.
!*!*...!*
!
21 kmmm
m
8
A little bit harder now
How many different ways to reorder the letters in the
word Statistics?
_ _ _ _ _ _ _ _ _ _
How many different ways to reorder the letters in the
word Probability?
400,50!2!3!3
!10
200,979,9!2!2
!11
9
Example (XVIII)
At an academic conference, 12 faculties are going to take
a picture
together. There are 3 professors, 5 associate professors
and 4
assistant professors. If we want people at the same level to
stay
together, how many ways to line them up?
10
Example (CLII)
My new bike lock has three dials numbered between 0 and
9.
How many different ways can the code be set if:
• No restrictions at all?
• None of the numbers may be the same?
• No two consecutive numbers may be the same?
• The third number must be lower than the second?
11
Example (XLII)
You are required to select a 6-character case-sensitive
password for
an online account. Each character could be upper-case or
lower-case letter or a number from 0 to 9.
• No restrictions at all?
• The first character can not be a number?
• The last four characters must all be different?
• There must be at least one capital letter and at least
one number?
12
Example (MD)
I want to have four friends over (five including myself) and
I want to make sure none of us bring the same type of
liquor. The VBS next to me sells 17 types of liquor. What’s
the probability no two people bring the same kind of
liquor?
13
