# Combinations generator

This combinations calculator generates all possible combinations of m elements from the set of n elements.

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#### Timur

Created: 2014-07-21 07:28:42, Last updated: 2021-10-27 15:13:54

For example, if you have a set from 3 elements, {A, B, C}, the all possible combinations of size 2 will be {A,B}, {A,C} and {B,C}. That is, combination here refers to the combination of n things taken m at a time without repetition. The total number of possible combinations, as shown in Combinatorics – combinations, arrangements and permutations, is
$C_{n}^m=\frac{n!}{m!(n-m)!}$.
To use the combinations generator below, you need to fill the set (by default it consists of A, B, C, D, and E elements), and enter combination size. All combinations will be generated using a lexicographic algorithm. The description of the algorithm used by the generator can be found below the calculator.

#### Set

Value
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#### Combinations generator algorithm

Combinations are generated in lexicographical order. The algorithm uses indexes of the elements of the set. Here is how it works on example:
Suppose we have a set of 5 elements with indexes 1 2 3 4 5 (starting from 1), and we need to generate all combinations of size m = 3.

First, we initialize the first combination of size m - with indexes in ascending order
1 2 3
Then we check the last element (i = 3). If its value is less than n - m + i, it is incremented by 1.
1 2 4
Again we check the last element, and since it is still less than n - m + i, it is incremented by 1.
1 2 5
Now it has the maximum allowed value: n - m + i = 5 - 3 + 3 = 5, so we move on to the previous element (i = 2).
If its value less than n - m + i, it is incremented by 1, and all following elements are set to value of their previous neighbor plus 1
1 (2+1)3 (3+1)4 = 1 3 4
Then we again start from the last element i = 3
1 3 5
Back to i = 2
1 4 5
Now it finally equals n - m + i = 5 - 3 + 2 = 4, so we can move to first element (i = 1)
(1+1)2 (2+1)3 (3+1)4 = 2 3 4
And then,
2 3 5
2 4 5
3 4 5 - and it is the last combination since all values are set to the maximum possible value of n - m + i.

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