Understanding the Difference Between Combinations and Permutations

If you're aiming for a 700 or higher on the GMAT, odds are you've put some time into combinations and permutations. By now, you should know the basic formulas:

C = n! / (k!)(n-k)!

P = n! / (n-k)!

If you learned permutations from my Total GMAT Math, the second formula may not look familiar. Indeed, I don't teach permutations with the formula: You simply don't need it. But to gain an intuitive understanding of how combinations and permutations differ, it's important to see the formulas side by side.

Order Matters

At a conceptual level, the difference between combinations and permutations is that, in permutations, order matters. If you want to know the number of ways a teacher can arrange 3 children from a class of 8 in the front row, that's a permutation. If you just want to know how many distinct groups of 3 she can select, that's combinations.

In the equations, that is represented by a k! in the denominator. Intuitively, that k! eliminates all of the permutations that are representing the same combination.

To see how it works, let's work through the examples above.

First: the number of ways a teacher can arrange 3 children from a class of 8 in the front row. Without the formula, you can recognize that there are 8 options for the first seat in the front row; once one child is seated, there are 7 options remaining for the second seat; then six options remain for the third seat. The answer is (8)(7)(6) = 336.

In the formula, it's 8! / (8-3)!, or 8!/5! = (8)(7)(6).

Now, the combinations version. How many distinct groups of 3 children can a teacher select from a pool of 8 children? Again, n=8 and k=3:

c = 8! / (3!)(8-3)! = 8! / (3!)(5!) = (8)(7)(6)/3! = (8)(7) = 56

Algebraically, the difference is that 3! in the denominator, which makes the number of combinations six times smaller.

The Underlying Concept

Every combination represents several permutations. For instance, three of the children, A, B, and C, make up one combination. However, those three children could be arranged in a variety of ways: ABC, ACB, CAB, etc.

That in itself is a permutations question: In how many ways can three children be arranged? The answer is 3! = 6. Sounds familiar? That's no coincidence. Essentially, we're finding the number of combinations by starting with the number of permutations, then dividing by the number of possible permutations of each subset.

(That's a mouthful, I know; it's worth reading over a couple of times.)

Why This is Important

The GMAT is really good at figuring out how to test your understanding of the concepts. Whether it's rates, systems of equations, or combinations, the toughest questions often don't work with the formulas you've learned.

Take this example of a combinations question:

A committee of 3 people is to be selected from a group of 8 people, which includes 4 married couples. If the committee cannot contain more than one member of any married couple, how many 3-person committees are possible?

No matter how hard you try, you can't plug that in to the combinations formula. There's no way to account for the "only one member of a married couple" provision using n or k.

Instead, start with the number of permutations. That you can do. For the first member of the committee, there are 8 possibilities. Once one member is selected, not only is that person eliminated, but his or her spouse is, as well. That leaves 6 possibilities for the second person. Then another person is eliminated, as is another spouse. Finally, there are 4 possibilities for the remaining spot.

So, there are (8)(6)(4) permutations, for a total of 192. That's not the answer, though it's likely to be one of the choices.

Since the committee consists of 3 members, there are 3! possible permutations of any committee. Thus, we need to divide the number of permutations by 6 to get the number of combinations:

(8)(6)(4) / 6 = 8(4) = 32

There's our answer. No matter how you approach it, it's not an easy question, but without an understanding of the concepts that the GMAT is testing, you probably don't have a chance.

Resources

If this is over your head, it's important to take a step back and understand the more basic concepts: the differences between combinations and permutations, and how to recognize them. There are two chapters in Total GMAT Math on these topics.

There are a number of permutations and combinations questions in The Official Guide to the GMAT, but not very many at this level. You'll find several difficult questions in this vein in my Extreme: Challenge set, as well as a few in Problem Solving: Challenge and Arithmetic: Challenge.

About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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