Aiming for a 700+ on the GMAT? You never know when a challenging combination or permutation question will pop up three-quarters of the way through your exam to wreck havoc on your score. This advanced concept is not as commonly tested as algebra fundamentals or number properties, but it’s definitely worth knowing the basics in case you do see it.
The Fundamental Counting Principle states that if an event has x possible outcomes and a different independent event has y possible outcomes, then there are xy possible ways the two events could occur together. For example, how many three-digit integers have either 6 or 9 in the tens digit and 1 in the units digit?
To solve, we need to find the possible outcomes for each digit (hundreds, tens, and units) and multiply them. Each digit has 10 possible values (0 through 9). The hundreds digit can be any of these except 0 (since a three-digit number cannot begin with 0). The tens digit has only 2 options (6 or 9). The units digit has only 1 possibility (1). Therefore, the total number of possibilities is 9 x 2 x 1 = 18.
Permutations are sequences. In a sequence, order is important. How many different ways can four people sit on a bench? For the first spot on the bench, we have 4 to choose from. For the next spot we’ll have 3, for the third spot we’ll have 2, and the last remaining person will take the final spot. Therefore, there are 4 x 3 x 2 x 1 = 24 ways. Harder permutations problems will require you to use this formula:
n = the number of options
r = the number chosen from those options
For example, how many possible options are there for the gold, silver, and bronze medals out of 12 athletes? Here n = 12 and r = 3. Since the order in which the athletes finish matters, we know to use the Permutation formula:
n! / (n – r)! = 12! / (12 – 3)! = 12! / 9! = 12 x 11 x 10 = 1,320 options
Combinations are groups. Order doesn’t matter. The Combination formula is only slightly different from the Permutation formula:
Let’s say Dominic took 10 photos. He wants to put 7 of them on Facebook. How many groups of photos are possible?
n! / r! (n – r)! = 10! / 7! (10 – 7)! = 10! / 7! 3!
= 10 x 9 x 8 / 3 x 2 x 1
= 720 / 6
= 120 different groups
Remember to ask yourself whether order matters in the problem, and don’t forget the Fundamental Counting Principle! The GMAT may also combine one or more of these concepts in a longer Word Problem to make the question more challenging, but if you can remember these basics, you’ll be good to go!
Vivian Kerr is a regular contributor to the Veritas Prep blog, providing tips and tricks to help students better prepare for the GMAT and the SAT.