When dealing with permutations and combinations, you are essentially trying to find the number of different outcomes given a set of items and a number of restrictions. The difference between permutation and combination merely depends on whether the order matters. Let me illustrate with an example. Suppose you have three food items, apples, bananas and carrots. If you were asked to only pick two of them, how many possibilities are there? If you got 3, that’s the right answer. You have apples & bananas, apples & carrots and bananas and carrots. This is combination. If I told you however, that the order in which you eat the food matters, then you have more possibilities, because instead of just apples & bananas, you have to consider bananas & apples too. The latter is permutation.
Listing
What I did earlier, when I listed out the 3 choices is called, not surprisingly, listing. But there is a method to listing to ensure that you don’t leave any possibilities out. Everyone tends to have his or her own method, let me share mine. If I have four items – a, b, c and d – and I’m supposed to choose 2:
Step 1: I take the first item a and combine it with b. Then I go down the list and combine it with c and d to get a total of three possibilities: ab, ac, ad.
Step 2: I take the second item b and combine it with the rest of the list. I don’t combine it with a because I already did that in step 1. So I get: bc, bd
Step 3: Keep doing the same thing for the next thing on the list, which is c. For c, there’s only one thing behind it on the list – d. So there’s only 1 possibility here: cd
Step 4: Keep going until you get to the second last item. In this case, we already reached it in step 3. Tally up your choices: ab, ac, ad, bc, bd, cd
Tree Diagram
If you noticed, listing is a good method for figuring out combinations. The Tree Diagram is a graphical method that helps you with permutation. Using the previous example of 4 items, here’s the tree diagram that illustrates the number of ways of permuting items.
If you look at the last line of the tree diagram and count the number of boxes, you will see that there are 12 possible ways to permute 2 items out of 4. Each item can be combined with 3 other things. Another way of seeing this is that you have two spaces you need to fill ___ and ___. Looking at the first space, you have 4 possible items you could place there. Looking at the second space, you have 3 possible items you could place there, since you have already put 1 item in the first space. So there are 4 x 3 = 12 ordered possibilities.
Fundamental Counting Principle
The spaces-and-number-of-options-to-fill-that-space method of thinking is essentially the fundamental counting principle. The principle states that
if task A can be done in m ways and after Task A is complete, Task B can be done in n ways, then there are m*n ways of completing Task A then Task B.
Let’s see if you understand the principle with a quick example. How many different ways are there are arranging these letters: T A N G O
Did you get 120?
Imagine five spaces ___ ___ ___ ___ ___
You have 5 possible letters to put in the first space, and when that’s done, you have 4 possible letters to put in the second space, 3 letters for the third space and so on. So the answer is 5*4*3*2*1 or 5!
If you’re comfortable using these methods to solve permutation and combination problems, stay tuned for the second installment that uses direct formulas to calculate the number of possibilities. In the meantime, check out Grockit for GMAT quantitative practice.
ompound Interest
= 3. Recognize that you are using a smaller number, so your result will be smaller too. Test makers love to give tempting answer choices that assume
=
+ (n – 1)d, where
= d
, where
/ 





, the equation
is satisfied. Pick an easy number to check, like 0.















and
have two factors of two (4) and two factors of three (9).
is also a perfect square, we can rewrite the expression:
, is
divisible by 4?
is divisible by 10
, we can quickly see that the three factors are consecutive numbers. In order for the product of three consecutive numbers to be divisible by 4, either one has to be divisible by 4 by itself, or two must be divisible by 2 individually. This means that both
and
must be even.
is divisible by 10, which means that one of the two factors is divisible by 5 and also that one of the two is divisible by 2. However, it does not tell us which of the two is divisible by 2, so we cannot know if the remaining
is odd. If