Download Counting Techniques

Introduction to
Probability & Statistics
Counting
Fundamental Rule
 If
an action can be performed in m ways and
another action can be performed in n ways,
then both actions can be performed in m•n
ways.
Fundamental Rule
 Ex:
A lottery game selects 3 numbers
between 1 and 5 where numbers can not be
selected more than once. If the game is truly
random and order is not important, how
many possible combinations of lottery
numbers are there?
Fundamental Rule

Ex: A lottery game selects 3 numbers between 1 and 5 where
numbers can not be selected more than once. If the game is
truly random and order is not important, how many possible
combinations of lottery numbers are there?
1
2
3
4
5
Fundamental Rule

Ex: A lottery game selects 3 numbers between 1 and 5 where
numbers can not be selected more than once. If the game is
truly random and order is not important, how many possible
combinations of lottery numbers are there?
1
2
3
4
5
2
3
4
5
Fundamental Rule

Ex: A lottery game selects 3 numbers between 1 and 5 where
numbers can not be selected more than once. If the game is
truly random and order is not important, how many possible
combinations of lottery numbers are there?
1
2
3
4
5
2
3
4
5
3
4
5
Fundamental Rule

Ex: A lottery game selects 3 numbers between 1 and 5 where
numbers can not be selected more than once. If the game is
truly random and order is not important, how many possible
combinations of lottery numbers are there?
1
2
3
4
5
2
3
4
5
3
4
5
LN = 5•4•3
= 60
Combinations
 Suppose
we flip a coin 3 times, how many
ways are there to get 2 heads?
Combinations
 Suppose
we flip a coin 3 times, how many
ways are there to get 2 heads?
Soln:
List all possibilities:
H,H,H
H,H,T
H,T,H
T,H,H
H,T,T
H,T,H
T,H,H
T,T,T
Combinations
Of 8 possible outcomes, 3 meet criteria
H,H,H
H,H,T
H,T,H
T,H,H
H,T,T
H,T,H
T,H,H
T,T,T
Combinations
If we don’t care in which order these 3 occur
H,H,T
H,T,H
T,H,H
Then we can count by combination.
3!
3  2 1

3
3 C2 
2 !(3  2)! 2  1 (1)
Combinations
 Combinations nCk
= the number of ways to
count k items out n total items order not
important.
n!
k Cn 
k !(n  k )!
n = total number of items
k = number of items pertaining to event A
Example
 How
many ways can we select a 4 person
committee from 10 students available?
Example
 How
many ways can we select a 4 person
committee from 10 students available?
No. Possible Committees =
10! 10  9  8  7  6!

 1,260
10 C4 
4! 6! 4  3  2  1 6!
Example
 We
have 20 students, 8 of whom are female
and 12 of whom are male. How many
committees of 5 students can be formed if we
require 2 female and 3 male?
Example
 We
have 20 students, 8 of whom are female
and 12 of whom are male. How many
committees of 5 students can be formed if we
require 2 female and 3 male?
Soln:
Compute how many 2 member female
committees we can have and how many 3
member male committees. Each female
committee can be combined with each male
committee.
Example
8!
12!

 6,160
8 C2 12 C3 
2! 6! 3! 9!
Permutations
 Permutations
is somewhat like combinations
except that order is important.
n!
n Pk 
(n  k )!
Example
 How
many ways can a four member
committee be formed from 10 students if the
first is President, second selected is Vice
President, 3rd is secretary and 4th is
treasurer?
Example

How many ways can a four member committee be
formed from 10 students if the first is President,
second selected is Vice President, 3rd is secretary and
4th is treasurer?
10!
 5,040
10 P4 
(10  4)!
Example

How many ways can a four member committee be
formed from 10 students if the first is President,
second selected is Vice President, 3rd is secretary and
4th is treasurer?
10P4
•
•
= 10*9*8*7 = 5,040