7. Calculating Permutations

Overview

Permutations tell us how many ways we can arrange a set of objects, and are used in many probability problems. Learn how to calculate the number of permutations a problem has in this lesson.

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Summary

  1. Lesson Goal (00:14)

    The goal of this lesson is to learn what permutations are and learn how to calculate them.

  2. Overview of Permutations (00:44)

    A permutation is an ordered arrangement of some or all of the items in a set. For example, if we have a series of 5 colored squares, any ordered arrangement of these squares represents a different permutation. 

    We may want to find the number of possible permutations for a set of items. To do this, we can consider the process of creating a new permutation. As an example, if we have 5 items, there are 5 possible items we can put in the first slot, 4 possible items for the second slot, and so on. As a result, the number of permutations is 5 x 4 x 3 x 2 x 1, which is 120. 

Transcript

Over the previous lessons, we've learned a lot about probability.

We're now going to look at a related concept, permutations.

In this lesson, we'll learn what permutations are and how to calculate them.

Permutations can be useful in many situations in probability and statistics, particularly when we want to select objects from a population.

To understand permutations, consider these five colored squares. As we can see, there are many ways in which these five squares can be arranged. Each of these arrangements is a permutation.

A permutation is an ordered arrangement of some or all of the items in a set.

The key thing to note is that the order of the items is important. In this case, we're using the same five colors each time, but the order of the colors changes, meaning each row is a different permutation.

Let's now consider how we can calculate the number of possible permutations for these five squares.

To do this, we'll consider the process of creating a new permutation for this set of five squares.

When deciding which squares should come first in the a new permutation, we have five possible options. Let's add one of these to our new permutation.

Now that we've made one choice, there are only four possible options for the second square in the permutation. Again, we'll pick one of these options and add it to the permutation.

This process continues for the rest of the permutation. For the third square, there are three possible options. For the fourth square, there are two possible options.

And for the final square, there is only one remaining option.

To calculate the number of possible permutations, we simply multiply the number of options for each slot in the permutation.

In our case, this is five times four times three times two times one, which is 120.

Therefore there are 120 ways of arranging this set of five squares.

Let's consider how the number of permutations is affected when the number of squares changes Here we can see that for three squares, there are only six permutations, but for seven squares, there are 5,040 permutations.

It's clear that as the number of objects in a permutation grows, the number of possible permutations increases rapidly.

In each case, the number of permutations is calculated by multiplying all the numbers from the starting point down to one. This calculation is called a factorial and we indicate it with an exclamation mark.

Here we would say that three factorial is six and seven factorial is 5,040.

Now that we understand permutations and factorials, let's apply them to a practical problem.

A company has seven employees and wants to randomly select three to serve on a new committee. We want to know how many possible committees can be picked. The first person picked will be the chairperson of the committee and the second person picks will be the secretary. So the order in which people are picked is important. As a result, this problem needs to be solved using permutations. Let's create a permutation for the committee.

There are seven possible people who can fill the first spot. There are then six remaining people who can fill a second spot. And five people who can fill the final spot.

As before, we multiply these numbers to find the total number of permutations, which is 210.

We use the notation 7P3, sometimes called seven pick three for this calculation. This tells us that there are 210 possible committees that we can pick from these seven people in the case where the order of selection is important.

As we can see, calculating permutations can be quite intuitive.

However, we'd like to have a general mathematical formula we can use to calculate permutations.

As we've seen, we can calculate 7P3 as seven times six times five.

Let's multiply this calculation by four by three by two by one and then divide by four by three by two by one.

Because we multiplied and divided by the same number, our calculation has the same value, but we can see that the numerator is now seven factorial and the denominator is four factorial.

We want to change the denominator so that it only uses the number seven and three, as these are the numbers in our 7P3 expression.

We'll change the four to seven minus three.

Now we can create a general permutations formula.

For the calculation and nPr, we divide N factorial by N minus R factorial.

nPr means that this formula tells us the number of permutations of our objects that we can create from a set of N objects. In many cases, the intuitive approach we saw earlier might be easier to understand, but there are some situations where it's useful to know that there is a formula for calculating permutations. This concludes our look at permutations. In the next lesson, we'll learn about the similar concept of combinations.