# Combination Formula

The combination formula describes the possible combination of elements out of a group of elements. In combinations, order does not matter. See the formula for permutations for where order does matter.

Expression | Description |
---|---|

The total number of elments in the group | |

The number of elements to select |

The combination formula describes the possible combination r elements out of a group of n elements where order does not matter.

To calculate the possible combinations of choosing two items from the set {♥︎,♦︎,♣︎,♠︎} we can set up the formula. Substitute into the equation which represents the size of the set and substitute into the formula which represents the number of items we are choosing.

Evaluate the subtraction.

Expand the factorial operator.

Simplify the fraction.

There are ways to select two items from the set {♥︎,♦︎,♣︎,♠︎}. This can be seen in the list of possible combinations shown below.

- { ♥︎, ♠︎ }
- { ♦︎, ♠︎ }
- { ♥︎, ♦︎ }
- { ♣︎, ♠︎ }
- { ♥︎, ♣︎ }
- { ♣︎, ♦︎ }

4 Choose 3

This example shows how to calculate the possible ways to select three items from the set of four represented as {♥︎,♦︎,♣︎,♠︎}.

Start by setting up the combination formula.

Substitute into the equation which represents the size of the set and substitute into the equation which represents the number of items we are choosing.

Evaluate the subtraction in the denominator.

Expand the factorial operator.

Simplify the fraction

There are possible ways to select three items from the set {♥︎,♦︎,♣︎,♠︎} shown in the list below.

- {♥︎,♦︎,♣︎}
- {♥︎,♦︎,♠︎}
- {♥︎,♣︎,♠︎}
- {♦︎,♣︎,♠︎}