Combinatorics. The generator of combinations.
Combinatorics. The generator of combinations of m from n.
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.
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.
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