Everyone knows add, subtract, multiply, and divide, but the factorial symbol is slightly more obscure. It will not come up very frequently on the GMAT math, but with the knowledge of just a few facts about factorials, you will be able to master this question.
The Factorial
When a mathematician writes 5! it may look like the mathematician is very excited about the number 5, but actually, this is mathematical notation with a specific meaning. It is read "five factorial," and it means: the product of all the positive integers from 5 down to one.
5! = (5)(4)(3)(2)(1) = 120
One trick to know is how to divide factorials. Suppose a GMAT Problem Solving question asks: "What is the value of (8!)/(6!)?" Whenever you divide a bigger factorial by a smaller factorial, the entire denominator, the smaller factorial, will cancel out. Think about it:
All six factors in the denominator are identical to the last six factors in the numerator, leaving only the first two factors of the numerator, so they cancel. This kind of cancellation will always happen when you divide a larger factorial by a smaller factorial.
Permutations
In advanced mathematics, factorials have myriad uses. In terms of GMAT math, the biggest application of them will be to permutations. A permutation is an ordered arrangement of a set. Suppose five students are going to sit in five seats: one could ask the question, how many different possible orders are there for the five children on the five seats? In other words, how many permutations are there?
The number of permutations with n distinct objects is n! If five children are going to sit on bench, or in a row of chairs, there are 5! = 120 different permutations possible.
Beware: the answer is different if the five seats are around a round table or the five chairs are arranged in a circle. Let's call the children A, B, C, D, and E. If they are sitting on bench, then ABCDE and CDEAB are quite different. If they are sitting at a round table, though, these two represent the same order, as seen in the diagram below:
In that situation, you just have to say that A sits somewhere, anywhere, and then you have a permutation of the other four students with respect to where A is. There are 4! = 24 different possible seating arrangements.
This post was written by Mike McGarry, GRE and GMAT Expert at Magoosh Test Prep. Magoosh offers hundreds of practice questions and video lessons, as well as free resources and tips on how to master the GRE and GMAT. For more math help, read our post on the Top 5 GMAT Math Formulas.
