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# Combinations

Probability and Statistics
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Probability Theory
schedule Jul 1, 2022
Last updated
local_offer Arthur
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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:

$${n \choose {k}} = \frac{n!}{k!\,(n-k)!}$$

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. $k = 2$).

\begin{align*} {5 \choose {2}} &= \frac{5!}{2!(5-2)!} \\ &= \frac{5!}{2!\,3!} \\ &= \frac{120}{2 * 6} \\ &= 10 \end{align*}

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".

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