Permutations and Combinations

Sep 15, 2026

Permutations and Combinations provides a structured overview of counting methods used when order matters or doesn’t, and when repetition is allowed or not. The material defines the variables n and r, then applies the relevant permutation or combination formulas to practical scenarios such as counting codes with repeated digits, ordering balls from a pool without repetition, and forming ice-cream scoop selections where flavors may repeat. Additional examples demonstrate committee selection, password construction with mixed letter and digit rules, and arranging multiple volumes using factorial reasoning.

LanguageEnglish

CategoryExam

FormatPDF

Pages2

Reading Time5 min

AccessFree to download

Permutations and Combinations

Types of formulas (with and without repetition)

Examples

Permutations and Combinations | 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 = 17,576 The different ways you can arrange the digits = = 1,000 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

FAQs

What do n and r represent in permutations and combinations problems?

n is the number of things available to choose from, and r is the number of things selected.

When should you use a permutation formula versus a combination formula?

Use permutations when order matters, and combinations when order doesn’t matter.

How do repetition rules change the counting method?

If repetition is allowed, use the formulas labeled with repetition; if repetition is not allowed, use the ones labeled without repetition.

Sep 15, 2026

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