# Trevor Klee, Tutor

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### What to know about combinations and permutations for GMAT quant

This is by far the most dreaded topic on the GMAT. Here’s what you need to know about it, excerpted from my guide to GMAT quant.

## How to recognize:

Any time a question asks about the number of different ways or combinations something can be arranged, or if we’re trying to count the ways we can pick a small group out of a bigger group.

## What to know:

1. The most important two words when asked to count things are AND and OR. If we’re combining things (AND), we multiply. If we’re picking options (OR), we add.

Example:

If I have a red, blue, and green ball in a bag, I have three ways of picking any ball out of the bag.

If I only want to pick a red ball OR a blue ball, I have two ways of doing that.

If I want to pick two balls (one ball, put it back, AND pick another ball), I have nine (3*3) ways of doing that. I could pick red then blue, red then green, or red then red. I could also start with blue or green, so, if you add it up, that’s 9 in total.

2. If we’re asked to order a bunch of things, that’s a permutation. In order to answer these, we need to do a factorial. If I have x things, then a factorial is x(x-1)(x-2)…

The notation for factorial is an exclamation point: ! . So 5!=5*4*3*2*1.

Example:

We’re asked how many different ways we could order 5 people around a dinner table. For the first spot, I have 5 options. Once I pick the first spot, I then go to the next spot. I have to pick out of 4 options for the next spot.

So, I have 5 options for the first spot AND 4 options for the second AND 3 options for the third AND 2 options for the fourth AND 1 option for the last spot. So my total options are 5*4*3*2*1=120.

Note that it’s similar logic if I’m ordering 5 people around a dinner table, but there are only 3 spots. I just stop my permutation earlier. I have 5 options for the first AND 4 options for the second AND 3 options for the third, so my total options are 5*4*3.

3. Sometimes, I might have repeats when ordering. In that case, I need to divide my permutation by a factorial of the number of repeats.

Example:
If I’m asked how many ways I can order the letters AAABBCC, I start off with the same factorial as before: 7 options for the first spot AND 6 for the second… etc.

Now, let’s pretend the first A is A1, the second A is A2, etc. It doesn’t affect the math, but it’ll make the following explanation easier.

The problem with a normal factorial this counts A1 A2 A3 B1 B2 C1 C2 as separate from A3 A2 A1 B2 B1 C2 C1, which it isn’t. It’s the same permutation: AABBCC .

So I need to divide out the number of repeats. How many repeats do I have? Well, I have 3 A’s that will go in 3 different spots, so in terms of repeats, I have 3 options for the first spot, 2 options for the second, 1 option for the third.

The same logic will go for my 2 B’s and my 2 C’s.

So, altogether, my equation is 7*6*5*4*3*2*1/((3*2*1)(2*1)(2*1). If I cross out my common factors, I get 7*6*5=210.

4. This brings us to combinations. Combinations are when you pick a smaller group out of a bigger group, like picking which 3 friends you are going to take to an amusement park out of a group of 8.

If the bigger group is n, and the smaller group is r, then this is normally written as nCr, and the formula is n!/(r!(n-r)!). The derivation of the formula is explained in the example below.

Example:

If we’re picking 3 friends out of 8 to go to the water park, this is essentially both the limited spots from our permutations and the repeated spots from my permutation.

The limited spots should make sense: I have 8 options for my first spot AND 7 options for my second AND 6 options for my third.

For the repeats, the logic is that picking my friends Jeff, Bob, and Larry is the same as picking my friends Bob, Larry, and Jeff.

So, I need to do my limited permutation, which is equivalent to 8!5!, because that will make sure I only multiply 876.

Then, I need to divide by my repeats, which are going to be equal to the factorial of the smaller group. To take my friends Jeff, Bob, and Larry, I have 3 ways of repeating the first spot, 2 ways of repeating the second, and 1 way of repeating the third.

So, I divide 8*7*6/3!. This gives me 56.

## Takeaways:

1. AND means multiply, OR means add .

2. Permutations mean factorial.

3. Repeats mean divide by the factorial of the number of repeats.

4. Combinations are n!/(r!(n-r)!).

If you want practice with this concept and detailed explanations, you should get my GMAT quant book.