Permutation Calculator – Calculate Permutations and Combinations

The permutation calculator is online tool designed to compute the number of ways to arrange a subset of items from a larger set, where the order of arrangement is significant.

When you have three different colored balls: red, blue, and green. The permutation calculator would help you determine all possible ways to arrange these balls in a line.

In this case:

• RGB
• RBG
• BRG
• BGR
• GRB
• GBR

The calculator would tell us there are 6 possible arrangements (3! = 3 × 2 × 1 = 6).

Permutation Calculator

Set Size (n)	Selection Size (r)	Permutation P(n,r)	Combination C(n,r)
5	2	20	10
6	3	120	20
7	4	840	35
8	3	336	56
9	5	15,120	126

Permutation Combination Calculation Formula

The formulas for permutations and combinations are:

Permutation Formula (with repetition allowed):

P(n,r) = n!/(n-r)!

Where:

• n = total number of items
• r = number of items being arranged
• ! = factorial

Combination Formula:

C(n,r) = n!/[r!(n-r)!]

Let’s calculate the number of ways to select 3 people from a group of 5 for a committee where order matters (permutation):

P(5,3) = 5!/(5-3)!
       = 5!/(2)!
       = 120/2
       = 60 possible arrangements

How to Calculate Permutation?

Calculating permutations involves these steps:

• Identify the total items (n)
• Determine items to be arranged (r)
• Apply the formula
• Simplify the expression

How many different 4-letter arrangements can be made from the word “MATH”?

• n = 4 (total letters)
• r = 4 (using all letters)
• P(4,4) = 4!/(4-4)!
• = 24/1
• = 24 possible arrangements

How many combinations with 3 numbers?

Using numbers 1-9, selecting 3 numbers:

C(9,3) = 9!/[3!(9-3)!]
       = 9!/(3! × 6!)
       = **84 combinations**

How many possible combinations of 4 numbers?

Using numbers 1-9, selecting 4 numbers:

C(9,4) = 9!/[4!(9-4)!]
       = 9!/(4! × 5!)
       = 126 combinations

How many combinations with 5 numbers?

Using numbers 1-9, selecting 5 numbers:

C(9,5) = 9!/[5!(9-5)!]
       = 9!/(5! × 4!)
       = 126 combinations

How many possible combinations of 2 numbers?

Using numbers 1-9, selecting 2 numbers:

C(9,2) = 9!/[2!(9-2)!]
       = 9!/(2! × 7!)
       = 36 combinations

Sources and References:

• Mathematical Association of America (MAA) – www.maa.org/math-topics/discrete-mathematics
• National Institute of Standards and Technology (NIST) – www.nist.gov/dads/HTML/permutation.html

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