Permutations and Combinations

-- Section 8.6 --

 
Multiplication Principle:  Let E1 and  E2 be two events. E1 can be performed in a1 ways.  E2 can be performed in a2 ways.  Event E1 followed by event E can be performed in   a1 a2  ways. 
Permutation:  An Ordered arrangement of objects
P(n,r)   = the number of permutations of n objects taken r at a time.
Combination: An Un-Ordered arrangement of objects
C(n,r)   = the number of combinations of n objects taken r at a time.

Click here for Factorial, Permutations and Combinations calculator
   
 Lets say 3 people (Andy, Betty, and Clark) compete in a running race.  How many different outcomes are possible?  
This a permutation because order is important.
There 6 different outcomes possible.  
1st Place 2nd Place 3rd Place
Andy Betty Clark
Andy Clark Betty
Betty Clark Andy
Betty Andy Clark
Clark Andy Betty
Clark Betty Andy

Solve

Step

          Spot Check

Permutation Formula
Take 3 objects from 3 objects

            P(n, n) = n!

You can also view this problem as 3! = 3 2 1 = 6.  That is, how many different ways can  3 objects be arrangement?

There are 3 choices for the first spot.  That leaves 2 choices left for second spot.  After the first 2 spots are chosen, there is 1 left for the 3rd spot.

  
  5 people are standing in a line.  How many different outcomes are possible?  

[Solution]

 In how many ways can you select a committee of 3 people from a group of 12 members?

This is a combination because the 3 person committee are not ordered.  
Solve Step

Combination Formula

You can cancel the 9!'s

C(12, 3) = 220

220 different 3 people committees

 

  In how many ways can you select a committee of 4 people from a group of 10 members?

[Solution]

In how many ways can you select a committee of 3 people from a group of 12 members? The committee members consist of a chairperson, treasurer and a secretary. 

This is a permutation because the 3 person committee is ordered because each person has a position.
For example:  
Andy - Chair, Betty - Treasurer, Clark - Secretary  is different from
Andy - Treasurer, Betty - Chair, Clark - Secretary  

Solve

Step

Permutation Formula
You can cancel the 9!'s

P(12, 3) = 1320

1320 different 3 person committees

In how many ways can you select a committee of 4 people from a group of 10 members? The committee members consist of a chairperson, treasurer, officer, and a secretary. 

[Solution]

A women has 4 blouses, 3 skirts and and 5 pairs of shoes.  Assuming the woman does not care what she looks like, how many different outfits can she wear?
Solve Step

4 3 5 = 60 

Use the Multiplication Principle - Click here

She has 4 3 5 = 60 different outfits.

License plates for cars have to be unique. If a license plate contains 6 characters consisting of 2 letters followed by 4 digits  example:  QW2354

  • how many different license are possible?

  • how many different license are possible if letters were allowed to repeated but numbers are not allowed to be repeated?

[Solution]

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada