Permutations and combinations

What is a permutation?

In order to answer many probability questions, we need to understand permutations and combinations.

A permutation is the number of ways you can arrange a set of things, and the order matters.

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The formula for a permutation is

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

where $n$ is the total number of items we have, and $k$ is the number of items we want to arrange.

The difference between permutations and combinations, and how to calculate both

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Permutation example problem

Example

I have $4$ scoops of ice cream: $1$ chocolate, $1$ strawberry, $1$ vanilla, and $1$ mint. I want to eat only $3$ of the scoops. How many different ways can I eat $3$ of the scoops if I consider both which scoops I eat and the order in which I eat them?

This is a permutation question, since I care about the order in which I eat the scoops. There are $4$ total scoops, but I only want to eat $3$ of them. Therefore, the number of ways I could eat three of the scoops of ice cream is

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

$_4P_3=\frac{4!}{(4-3)!}$

$_4P_3=\frac{4!}{1!}$

$_4P_3=\frac{4\cdot3\cdot2\cdot1}{1}$

$_4P_3=4\cdot3\cdot2$

$_4P_3=24$

There are $24$ different ways that I could eat $3$ of the $4$ scoops. For example, chocolate-strawberry-vanilla would be $1$ of the $24$ options, but since order matters, chocolate-vanilla-strawberry would be another option.

On the other hand, a combination is the number of ways you can arrange a set of things, but the order doesn’t matter. The formula for a combination is

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

where $n$ is the total number of items we have, and $k$ is the number of items we want to choose. Sometimes people write $_nC_k$ as

$\binom{n}{k}$

which is called the binomial coefficient, and read as “ $n$ choose $k$ .”

So to continue with the example from earlier, chocolate-strawberry-vanilla and chocolate-vanilla-strawberry would not count separately, because we don’t care about the order when we’re talking about combinations. All we care about is which items we picked, so chocolate-strawberry-vanilla and chocolate-vanilla-strawberry would count as the same thing.

Permutations and combinations for Probability and Statistics.jpg — On the other hand, a **combination** is the number of ways you can arrange a set of things, but the order doesn’t matter.

Example

I have the same $4$ scoops of ice cream: $1$ chocolate, $1$ strawberry, $1$ vanilla, and $1$ mint. I want to eat only $3$ of the scoops, and I don’t care about the order in which I eat my $3$ scoops. How many different combinations of $3$ scoops can I create?

This is a combination question, since I don’t care about the order in which I eat the scoops. There are $4$ total scoops, but I only want to eat $3$ of them. Therefore, the number of ways I could eat three of the scoops of ice cream is

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

$_4C_3=\frac{4!}{3!(4-3)!}$

$_4C_3=\frac{4!}{(3!)(1!)}$

$_4C_3=\frac{4\cdot3\cdot2\cdot1}{(3\cdot2\cdot1)(1)}$

$_4C_3=\frac{4}{1}$

$_4C_3=4$

There are $4$ different ways that I could eat $3$ of the $4$ scoops. For example, chocolate-strawberry-vanilla would be $1$ of the $4$ options, but since order doesn’t matter, chocolate-vanilla-strawberry would be the same combination, and wouldn’t count as another one of the $4$ combinations.