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.