The GMAT quantitative (Maths) section often has questions involving concepts of Combinations and/or Probability. The GMAT preparation series on MBA Crystal Ball moves on, as the GoGMAT team covers questions that cover both Combinations and Probability .

### GMAT Maths: Combinations and Probability

The GMAT exam is designed to surprise you. There are no books detailing all the exact questions you are likely to face, and there is nobody who can walk into a test center with full confidence of scoring 800.

Some GMAT math questions are straightforward, and test a single math concept. However, harder questions, and those are the ones that contribute the most to your score, tend to mix different math topics. Questions that mix Combinations and Probability are a great example. There are almost unlimited examples; however, once you learn the core principles, you should be able to understand quickly what the question requires.

Unfortunately, these types of questions also seem to be the wordiest. We have to learn to step back from the wording and identify the important information fast. Thankfully this is easier then it looks when you have good methodology.

Let’s begin by reminding ourselves of the formulas for probability and combinations/permutations.

The formula for simple probability:

Probability of A happening = (number of times A can occur) / ( number of time any outcome can occur)

The formula for Permutations:

Permutations   =   n! / (n-r)!

And the formula for Combinations:

Combinations   =    n! / (r! (n-r)!)

Remember: in Permutations, the order matters. In Combinations—it does not.

Methodology:

There are two steps that will help you deal with GMAT probability and combinations/permutations questions efficiently.

1)      Work out whether the question asks for Combinations or Permutations

2)      Write out the equation and fill in what’s needed.

You simply dig into the wording and take out the information necessary to get to the answer fast.

Let’s look at an example:

Two dice are rolled. What is the probability that the sum will be greater than 10?

At the first glance, we can see that this question involves probability and calculating the number of ways in which any number can be rolled. So let’s look at the first step in our methodology:

1)      Work out whether the question asks for Combinations or Permutations

In this question the order of the numbers rolled is important. If we roll a 6 on die one, and a 4 on die two, then that is different from a 4 on die one and a 6 on die two; they are two different results, which naturally affects our probability calculation. Therefore, this question deals with permutations.

Working out the permutations of numerous similar objects such as dice is straightforward and does not require the use of a formula.

There are six different possibilities on a single die, as we can roll either a 1, 2, 3, 4, 5 or 6. If we have two dice, then the number of permutations is simply:

6 possible outcomes on die one  X  6 possible outcomes on die two

or

6  X  6 = 36 total permutations

Now we can move to step 2:

2)      Write out the equation and fill in what’s needed.

Our probability equation will look like:

Probability of rolling greater than 10 = (number of outcomes over 10) / ( total number of outcomes )

So far, we only know the denominator as we worked out the total number of permutations in step one. We now need to work out the number of times we can roll a number over ten. This step requires some basic maths and logical thinking. If we need to roll over ten with just two dice, then the number rolled must be either 11 or 12. So now, we need to calculate the number of ways in which we can roll 11 or 12.

We can do this the long way by calculating every permutation and it’s value:

 1, 1 = 2 1, 2 = 3 1, 3 = 4 1, 4 = 5 1, 5 = 6 1, 6 = 7 2, 1 = 3 2, 2 = 4 2, 3 = 5 2, 4 = 6 2, 5 = 7 2, 6 = 8 3, 1 = 4 3, 2 = 5 3, 3 = 6 3, 4 = 7 3, 5 = 8 3, 6 = 9 4, 1 = 5 4, 2 = 6 4, 3 = 7 4, 4 = 8 4, 5 = 9 4, 6 = 10 5, 1 = 6 5, 2 = 7 5, 3 = 8 5, 4 = 9 5, 5 = 10 5, 6 = 11 6, 1 = 7 6, 2 = 8 6, 3 = 9 6, 4 = 10 6, 5 = 11 6, 6 = 12

Or we can think about it logically. Rolling an 11 or 12 can be done in the following ways:

5, 6 = 11

6, 5 = 11

6, 6 = 12

That gives us three different ways that we can roll an 11 or 12. (The full list of possible outcomes was deliberately listed above so you can scroll through it and understand how they were calculated).

We can now add these figures into our equation to work out the answer:

Probability of rolling greater then 10= 3 / 36 = 1 / 12

GMAT probability + permutations questions generally require a little extra thinking. You have to think logically, divide the problem into several steps, and calculate the necessary parts of the equation.

The two general steps given above are a great guideline. Firstly, work out how many total options are available (this may require thinking about whether the question relates to permutations or combinations). Secondly, and perhaps most importantly, always write out the equation so you know which numbers you need to identify. This will guide you and focus your attention on the most important numbers.

Good luck!

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