# Combinatorics – combinations, arrangements and permutations

This calculator calculates the number of combinations, arrangements and permutations for given n and m

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Created: 2011-07-21 15:29:32, Last updated: 2020-12-16 10:38:25

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Below is a calculator that computes the number of combinations, arrangements and permutations for given n and m. A little reminder on those is below the calculator.

#### Combinatorics. Combinations, arrangements and permutations

Number of permutations from n

Number of arrangements of m from n

Number of combinations of m from n with repetitions

Number of combinations of m from n

So, assume we have a set of n elements.

Each ordered set of n is called a permutation.

For example, we have a set of three elements – А, В and С.
An example of an ordered set (one permutation) is СВА.
The number of permutations from n is
$P_n = n!$

Example: For the set of А, В and С, the number of permutations is 3! = 6. Permutations: АВС, АСВ, ВАС, ВСА, САВ, СВА

If we choose m elements from n in a certain order, it is an arrangement.

For example, the arrangement of 2 from 3 is АВ, and ВА is the other arrangement. The number of arrangements of m from n is
$A_{n}^m=\frac{n!}{(n-m)!}$

Example: For the set of А, В and С, the number of arrangements of 2 from 3 is 3!/1! = 6.
Arrangements: АВ, ВА, АС, СА, ВС, СВ

If we choose m elements from n without any order, it is a combination.

For example, the combination of 2 from 3 is АВ. The number of combinations of m from n is
$C_{n}^m=\frac{n!}{m!(n-m)!}$

Example: For the set of А, В and С, the number of combinations of 2 from 3 is 3!/(2!*1!) = 3.
Combinations: АВ, АС, СВ

Here is the dependency between permutations, combinations and arrangements
$C_{n}^m=\frac{A_{n}^m}{P_m}$
Note $P_m$ – the number of permutations from m

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