ProbabilityPermutations and Combinations

Permutations are known as any arrangement of distinct objects in a particular _________.

Permutations

order

A doctor has six examination rooms. There are six patients in a waiting room. In how many different ways can the patients be assigned to the examination rooms?

Example 1

Room 1 Room 2 Room 3 Room 4 Room 5 Room 6

6 5 4 3 2 1

=6 • 5 • 4 • 3 • 2 • 1

=720

A ______ is used to represent a factorial. A factorial is a type of repeated ________________.

For example.

Factorial !

multiplication

5! =5 • 4 • 3 • 2 • 1

=120

Permutations are solved using the formula

Like any formula we will use we need to know what the variables are!

n represents the total number of things r represents the number taken at a time

Permutation Notation

nP

r=

n!

n−r( )!

A baseball scout has received a list of 15 promising prospects. The scout is asked to list, in order of preference, the five most outstanding of these prospects. In how many different ways can the scout select the five best players?

Example 2

Total Number (n)

15 5Number Chosen (r)

=360360

How many different permutations are there of the letters in the words?

a) MATHEMATICS

Example 3

Total Number (n)

11 11Number Chosen (r)

=39916800

How many different permutations are there of the letters in the words?

b) MISSISSIPI

Example 3b

Total Number (n)

10 10Number Chosen (r)

=3628800

Combinations are a collection of distinct objects where ________ is _______ important.

The number of combinations of things taken at a time where order is not important is denoted:

Combinations

order not

nC

r=

n!

r ! n−r( )!

How many different 11-member football teams can be formed from a possible 20 players if any player can play any position?

Example 4

Total Number (n) Number Chosen (r)

20 11

20

C11

=20 !

11! 20 −11( )!=

20 !

11! 9( )! =167960

How many different poker hands consisting of five cards can be dealt from a deck of 52 cards?

Example 5

Total Number (n) Number Chosen (r)

52 5

52

C5=

52 !

5 ! 52 −5( )!=

52 !

5 ! 47( )! =2598960

What is the probability of being dealt a royal flush in five-card poker?

4 ways to draw with 4 suits

Example 5b

52

C5=

52 !

5 ! 52 −5( )!=

52 !

5 ! 47( )! =2598960

=

4

2598960

John has ten single dollar bills of which three are counterfeit. If he selects four of them at random, what is the probability of getting two good bills and two counterfeit bills?

We need to figure out a few different things to set up a full probability!

Example 6

Start with how many ways you can choose 4 bills from a possible 10.

Example 6

10

C4=

10 !

4 ! 10 −4( )!=

10 !

4 ! 6( )! =210

Next we need how many ways 2 cards can be chosen from the 7 good cards.

Example 6

7C

2=

7 !

2 ! 7 −2( )!=

7 !

2 ! 5( )! =21

Finally, we need to know how 2 counterfeit cards can be selected from 3.

Example 6

3C

2=

3 !

2 ! 3 −2( )!=

3 !

2 ! 1( )! =3

Now we put it all together!

Example 6

3C

2=

3 !

2 ! 3 −2( )!=

3 !

2 ! 1( )! =3

10

C4=

10 !

4 ! 10 −4( )!=

10 !

4 ! 6( )! =210

7C

2=

7 !

2 ! 7 −2( )!=

7 !

2 ! 5( )! =21

P = 7C

2•

3C

2

10C

4

P =

21• 3

210

P =

3

10