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:[latex]C_{n}^{k}=\dfrac {n!} {k!\left( n−k\right) !}[/latex]
For the number of combinations with repetition or reduction, we use this formula:
[latex]K_{n}^{k}=\dfrac {(n+k−1)!} {k!\left( n−1\right) !}[/latex]
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:[latex]C_{4}^{2}=\dfrac {4!} {2!\left( 4−2\right) !}\space =\space \dfrac{24}{2×2}\space =\space 6[/latex]
If there is repetition or reduction, we use this formula:
[latex]K_{4}^{2}=\dfrac {(4+2−1)!} {2!\left( 4−1\right) !}\space =\space \dfrac{120}{12}\space =\space 10[/latex]