Permutations are a type of combinatorics task that is often tested on the GMAT. In a permutations question, you are asked to find the number of ways to order a group of elements. The order of the elements matters in permutations questions, so the number of possible permutations can be quite large.
The permutations task is “lining up” a subgroup of elements spatially or temporally.
In our introduction, we learned that it makes sense to think of performing the permutations task as performing the combinations and ordering tasks successively: counting ways to select a subgroup and then counting ways to order a selected subgroup. However, this understanding does not link up with the standard formula for permutations – at least not at first glance.
Permutations as Ordering with a Stopping Point
To understand the standard formula for permutations, it’s better to think of permutations as “ordering with a stopping point.” Consider the following example:
Example: Lining Up Cars
How many ways can 4 cars from a collection of 7 cars be lined up in 4 parking spaces?
Let’s represent this scenario using the slot system from before:
___ ___ ___ ___
Since we are filling only four parking spaces, we will only use four slots. However, we still have seven options for the first slot!
___ ___ ___ ___
7
And we still have six options for the second slot, and so on and so forth:
___ ___ ___ ___
7 6 5 4
Once you’ve assigned these four cars you were asked to assign, you’re done. The number of ways to line up 4 cars from a collection of 7 cars in 4 parking spaces is 7 * 6 * 5 * 4 = 840
Permutations is like ordering with a stopping point, because we simply stop after assigning however many elements we were asked to assign.
The Standard Formula for Permutations
Now let’s talk about the standard formula for permutations. The full number of elements from which you select your subgroup is always represented as n. In this example with the cars, n = 7. But depending on where in the world you learn math, the number of elements in the subgroup may be represented as r or k. We’ll stick with r in these articles.
Here’s the standard permutations formula for ordering r things from a group of n things:
n!(n-r)!
Since n is the number of things in the whole group, and r is the number of things in the selected group, (n – r) is the number of things not being selected. We begin with the n! from the ordering task, and then we divide this number by the number of ways to arrange the things not being selected: (n – r)!
Connecting the Formula to the Slot Filling Method
Let’s go back to the car example. We were asked to line up 4 cars from a collection of 7 cars, so n = 7 and r = 4. In this case, (n – r)! = (7 – 4)! = 3! Here’s how this connects to our “slot filling” method:
___ ___ ___ ___ ___ ___ ___
7 6 5 4 3 2 1
The numbers after our stopping point correspond to the (n – r)! we divided out from the initial value of n! So there you have it: the permutations task is ordering with a stopping point.
If you are planning to take the GMAT, and need guidance through your journey, schedule a free consultation call with a top-scoring tutor to get your best score. Get a customized study plan and receive one-on-one instruction.
Contributor: Elijah Mize (Apex Instructor)