# Combination

## Combination

In a set E that includes n elements, any subset of E that includes k elements.

In a combination, the order of elements is not involved.

• combination without repetition or without reduction
Synonym of combination.
• combination with repetition or with reduction
Combination of elements in a set E in which the repetitions (or reductions) are allowed or where the order of elements chosen is not involved.

### Formula

The number of combinations of n elements of a set E taken k at a time is given by this relation:

$$C_{n}^{k}=\dfrac {n!} {k!\left( n−k\right) !}$$
For the number of combinations with repetition or reduction, we use this formula:
$$K_{n}^{k}=\dfrac {(n+k−1)!} {k!\left( n−1\right) !}$$

### Examples

Consider the set E = {2, 4, 6, 8}.

Here are a few examples of combinations of elements of E taken 2 at a time:

{2, 4}, {2, 8}, {6, 8}, {4, 8}.

Here are a few examples of combinations of elements of E taken 2 at a time with repetitions:

{2, 4}, {2, 2}, {6, 8}, {4, 4}.

The subsets {2, 8} and {8, 2} represent the same combination.

To calculate the number of combinations of elements of E taken 2 at a time, we use the formula:

$$C_{4}^{2}=\dfrac {4!} {2!\left( 4−2\right) !}\space =\space \dfrac{24}{2×2}\space =\space 6$$

If there is repetition or reduction, we use this formula:

$$K_{4}^{2}=\dfrac {(4+2−1)!} {2!\left( 4−1\right) !}\space =\space \dfrac{120}{12}\space =\space 10$$