*chevron_left*Probability Theory

# Combinations

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*schedule*Jul 1, 2022

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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

Here "apple & orange" is treated the same as "orange & apple", which is what we mean when we say "order does not matter".