# Permutations (Ordered Sampling Without Replacement)

An ordered set is a set where the order of the elements matter. Without replacement means that you can’t pick the same element more than once.

Theory

### Permutations

When you draw $r$ elements from a set of $n$ elements, you call the number of possible orders the permutations.

 $nP\phantom{\rule{-0.17em}{0ex}}r=\frac{n!}{\left(n-r\right)!}$

Note! The order of the drawn elements matters!

Example 1

15 teams are joining a relay race.

1.
How many different ways can you fill the three first places?
2.
Messi, Bale and Ronaldo each run for their team. What’s the probability that Bale’s team wins, Messi’s team comes in second, and Ronaldo’s team comes in third?

1.
Since it matters whether you get the first, second or third place, the order matters. That means you can think of the different ways to distribute the places as permutations. The calculation becomes
 $15P\phantom{\rule{-0.17em}{0ex}}3=\frac{15!}{\left(15-3\right)!}=2730.$

That means there are 2730 different ways to fill the three first places when there are 15 teams.

2.
Since the placement is decided, “Bale, Messi and Ronaldo” is one event out of the possible options. Then you get:

Example 2

You’re electing 4 students for the student council from a group of 31 people: One president, one vice president, one secretary and one treasurer.

1.
How many different councils can there be, when you take who gets which position into account?
2.
Al Gore is in the group. What is the probability that he is elected to the student council?

1.
Since it matters who has the different positions, you can think of this as a case where the order matters. That means each possible student council is a permutation. There are 31 students that can be chosen for president, 30 for vice president, and so on. Because you pick 4 council members out of the group, you find the number of different councils like this:
 $31P\phantom{\rule{-0.17em}{0ex}}4=\frac{31!}{\left(31-4\right)!}=755\phantom{\rule{0.17em}{0ex}}160$

That means you can elect $755\phantom{\rule{0.17em}{0ex}}160$ different student councils.

2.
As he’s one of the 31 students, and you assume that there’s an equal possibility for everyone to be elected, you have that: