Combinations can be described as the number of ways we can pick a set of $k$ elements from a total $n$ elements. Unlike for permutations, we are not interested in the order of elements, only whether an element is included in the $k$ elements or not.

The formula for combinations is:

where:

$n$: total number of elements

$k$: number of elements we want to pick

Example

How many different combinations of 2 fruits can we have from the following list of 5 (apple, orange, banana, grape, strawberry)?

Here the total number of elements / fruits is 5 (i.e. $n=5$).

We are interested in picking sets of 2 elements (i.e. $k = 2$).

We have 10 different possible combinations:

• apple & orange

• apple & banana

• apple & grape

• apple & strawberry

• orange & banana

• orange & grape

• orange & strawberry

• banana & grape

• banana & strawberry

• grape & strawberry