Transcript Permutation - Rice University

Permutation
A permutation is an arrangement in which
order matters.
A B C differs from B C A
How Many Permutations?
Consider four objects {A,B,C,D}
There are 4 choices for the first slot.
There are 3 choices for the second slot.
There are 2 choices for the third slot.
There is 1 choice for the last slot.
4 x 3 x 2 x 1 = 24 Permutations
ABCD
ACDB
BACD
BCDA
CABD
CBDA
DABC
DBCA
ABDC
ADBC
BADC
BDAC
CADB
CDAB
DACB
DCAB
ACBD
ADCB
BCAD
BDCA
CBAD
CDBA
DBAC
DCBA
Generalization
There are 4! ways to
arrange 4 items.
There are n! ways to
arrange n items.
Permutation Formula
In how many ways may r items be
selected out of a set of n items where
order matters
Permutation Example
Selecting 3 items out of a set of 5
We have 5 choices for the first item.
We have 4 choices for the second item.
We have 2 choices for the third item.
5 x 4 x 3 = 60 Permutations
Calculations
5  4  3  2 1 5!
5 43 

2 1
2!
5!
5 C3  C (5,3) 
(5  3)!
Permutation Formula
n!
n Pr  P ( n, r ) 
(n  r )!
Combinations
Combinations are arrangements in which
order does NOT matter.
A, B, C is the same as B, C, A
Evaluating
n Cr
In how many ways may 3 items be
selected from a set of 5 without regard to
order?
We already know that there 60
permutations of these items.
For each set of three, there are
3! or 6 arrangements.
ABC
BCA
ACB BAC
CAB CBA
All of these are really the same.
Our actual answer is 10.
5 C3
60
  10
3!
6
Consider the set {A,B,C,D,E}
These are the combinations.
A,B,C
A,D,E
A,B,D A,B,E A,C,D A,C,E
B,C,D B,C,E B,D,E C,D,E
Combination Formula
n!
n C r  C ( n, r ) 
r!(n  r )!
Lottery Calculations
In a lottery, 5 winning numbers are
selected out of a set of 15 numbers.
How many possibilities?
15!
 3003
15 C5 
5!10!
What is your probability?
You are likely to win one game
out of 3003 games.
Hot Diggety Dog!
Selecting Exactly 4 Winners
(This actually means 4 of the 5 correct numbers
and one of the 15 incorrect numbers.)
5 C4 15 C1
15 C5
5 15
75
25



3003 3003 1001
Selecting Exactly 3 Winners
Here is your opportunity to shine.
Calculate it yourself!
Check Your Answer
5 C3 15 C2
15 C5
10 105 1050 50



3003 3003 143
If you got this answer, smile and
pat yourself on the back.
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