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Permutations and Combinations Cheat Sheet
Type Formulas Explanation of Variables Example
Permutation with
repetition
(Use permutation formulas when order matters in the problem.)
Where n is the number of things to choose from, and you choose r of them.
A lock has a 5 digit code. Each digit is chosen from 0- 9 , and a digit can be repeated. How many different codes can you have?
n = 10, r = 5 105 = 100,000 codes
Permutation without
repetition
(Use permutation formulas when order matters in the problem.)
( )
Where n is the number of things to choose from, and you choose r of them. Sometimes you can see the following
notation for the same concept:
How many ways can you order 3 out of 16 different pool balls?
n = 16, r = 3
( )
Combination with
repetition
(Use combination formulas when order doesn’t matter in the problem.)
( ) ( )
Where n is the number of things to choose from, and you choose r of them.
If there are 5 flavors of ice cream and you can have 3 scoops of ice cream, how many combinations can you have? You can repeat flavors.
n = 5, r = 3 ( ) ( )
Combination
without repetition
(Use combination formulas when order doesn’t matter in the problem.)
( )
Where n is the number of things to choose from, and you choose r of them. Sometimes you can see the following notation for the same concept:
The state lottery chooses 6 different numbers between 1 and 50 to determine the winning numbers. How many combinations are possible?
n = 50, r = 6
( )
Permutations and Combinations
Examples
1) Mr. Smith is the chair of a committee. How many ways can a committee of 4 be chosen from 9 people given that Mr. Smith must be one of the people selected?
Mr. Smith is already chosen, so we need to choose another 3 from 8 people. In choosing a committee, order doesn't matter, so we need the combination without repetition formula.
( ) =^ ( ) = 56 ways
2) A certain password consists of 3 different letters of the alphabet where each letter is used only once. How many different possible passwords are there?
Order does matter in a password, and the problem specifies that you cannot repeat letters. So, you need a permutations without repetitions formula. The number of permutations of 3 letters chosen from 26 is
( ) =^ ( ) =^ 15,600 passwords
3) A password consists of 3 letters of the alphabet followed by 3 digits chosen from 0 to 9. Repeats are allowed. How many different possible passwords are there?
Order does matter in a password, and the problem specifies that you can repeat letters. So, you need a permutations with repetitions formula.
The different ways you can arrange the letters = 17, The different ways you can arrange the digits = = 1, So the number of possible passwords = 17,576 × 1,000 = 17,576,000 passwords
4) An encyclopedia has 6 volumes. In how many ways can the 6 volumes be placed on the shelf?
This problem doesn’t require a formula from the chart. Imagine that there are 6 spots on the shelf. Place the volumes one by one.
The first volume to be placed could go in any 1 of the 6 spots. The second volume to be placed could then go in any 1 of the 5 remaining spots, and so on. So the total number of ways the 6 volumes could be placed is 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 ways
Permutations and Combinations
