Combinatorics. The generator of combinations.

This calculator which generates possible combinations of m elements from the set of element with size n. Number of possible combinations, as shown in Combinatorics – combinations, arrangements and permutations is

The description of the generator algorithm is below the calculator.

Combinatorics. Generator of combinations.

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Combination size m

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Algorithm

Combinations are generated in lexicographical order. Algorithms use 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 combination size m = 3.
First, we initialize the first combination size m - indexes in ascending order
1 2 3
We start by checking the last element (i = 3). If its value 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 - last combination since all values are set to the maximum possible values of n - m + i.