The Permutation Formula explained

By Brent Hanneson - November 15, 2022

In a recent article, I noted that the GMAT doesn’t require students to know the permutation formula. I also stated that, if you understand how/why the permutation formula works, then you’ll understand why it’s unnecessary to memorize. So, let’s understand how/why the formula works.

The permutation formula goes something like this: 

10 Perm graphic 00.png

We’ll work up to this formula by examining two specific examples and then a general example. 

First question: 

How many arrangements of 3 objects are possible if the 3 objects must be chosen from 8 unique objects?

Let’s use the Fundamental Counting Principle (aka FCP, aka multiplication principle, aka slot method) to solve the question. 

There are 8 options for the first object in the arrangement and, to avoid duplication, there are 7 options remaining for the second object, and there are 6 options remaining for the third object. 

By the FCP, the total number of 3-object arrangements = (8)(7)(6) 

Note: (8)(7)(6) is the product of first 3 values of 8!.

Let’s apply the same strategy to solve this next question: 

How many arrangements of 4 objects are possible if the 4 objects must be chosen from 10 unique objects?

In this case, there are 10 options for the first object in the arrangement, 9 options for the second object, 8 options for the third object, and 7 options for the fourth object. 

By the FCP, the total number of 4-object arrangements = (10)(9)(8)(7) 

Note: (10)(9)(8)(7) is the product of first 4 values of 10!.

And now a generic version of the question: 

How many arrangements of r objects are possible if the r objects must be chosen from n unique objects?

This time there are n options for the first object, n-1 options for the second object, n-2 options for the third object, n-3 options for the fourth object, . . . and so on until we have selected the required r objects. 

By the FCP, the total number of r-object arrangements = (n)(n-1)(n-2)(n-3)….[until we’ve listed r values]

In general, we can write:

10 Perm graphic 01.png

As you can see, this “formula” is just a special case of the FCP, and, in a perfect world, we’d leave the formula exactly as shown above, since it provides a clear picture of the FCP at work.

The problem is mathematicians don’t like clunky formulas such as mine, especially if those formulas can be expressed more elegantly. So, here’s how we can express the product of the first r values of n! more elegantly:

First, in order to isolate the first r values of n!, we must eliminate the last n-r values (in blue):

10 Perm graphic 03 - 125 mag.png

We can do this by dividing n! by the product of the last n-r values like so: 

10 Perm graphic 04.png

Finally, we must recognize that another way to express the last n-r values of n! is (n-r)!.

This is the same logic we’d used to conclude that 5! represents the last 5 values of 8!, and that 3! represents the last 3 values of 7! 

Since (n-r)! is another way to express the last n-r values of n!, we can replace the denominator with (n-r)! to get: 

10 Perm graphic 05.png

In other words: 

10 Perm graphic 06.png

Now that you can see the permutation formula is really just a special case of the FCP, there’s really no reason to keep the formula in your mathematical toolbox.