# Combinations generator

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

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

.

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.

#### 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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