# Factorials - GMAT Math Study Guide

## Table of Contents

## Definitions

x factorial = x! = x(x-1)...(3)(2)(1)

- Factorial - Denoted by x!, the product of all positive integers less than or equal to x

For example:

2! = 2*1 = 2

3! = 3*2*1 = 6

5! = 5*4*3*2*1 = 120

10! = 10*9*8*7*6*5*4*3*2*1 = 3,628,800! - 0! = 1

By mathematical convention, zero factorial [denoted 0!] is 1.

## Uses of Factorials

Although the use of factorials will be covered in considerably greater detail in the permutations and combinations study guides, the following represents a brief example of the relevancy of factorials in solving word problems.

Three different letters, denoted A, B, and C, lie within a box. An individual will draw from the box three times. How many different ways can the letters be drawn?

The first drawing can turn out three ways as no letter has yet been drawn

The second drawing can turn out two ways as one letter has been drawn, meaning two letters remain in the box

The third drawing can turn out one way as two letters have been drawn, meaning one letter remains in the box

Translating this into algebra: 3*2*1 = 3! = 6 different ways (or orders) the letters could be drawn in.

Here are the six ways:

ABC

ACB

BAC

BCA

CAB

CBA

(If this does not make sense, please see the combinations page, which contains a much more thorough explanation.)

The first drawing can turn out three ways as no letter has yet been drawn

The second drawing can turn out two ways as one letter has been drawn, meaning two letters remain in the box

The third drawing can turn out one way as two letters have been drawn, meaning one letter remains in the box

Translating this into algebra: 3*2*1 = 3! = 6 different ways (or orders) the letters could be drawn in.

Here are the six ways:

ABC

ACB

BAC

BCA

CAB

CBA

(If this does not make sense, please see the combinations page, which contains a much more thorough explanation.)

Another Example:

How many different ways can five sports teams be ranked?

There are five different possibilities for first place as no team is yet ranked

There are four different possibilities for second place as one team is already ranked

There are three different possibilities for third place as two teams are already ranked

There are two different possibilities for fourth place as three teams are already ranked

There is one different possibility for fifth place as four teams are already ranked

Translating this to algebra: 5 * 4 * 3 * 2 * 1 = 5! = 120 different ways exist for the five teams to be ranked.

(If this does not make sense, please see the combinations page, which contains a much more thorough explanation.)

There are five different possibilities for first place as no team is yet ranked

There are four different possibilities for second place as one team is already ranked

There are three different possibilities for third place as two teams are already ranked

There are two different possibilities for fourth place as three teams are already ranked

There is one different possibility for fifth place as four teams are already ranked

Translating this to algebra: 5 * 4 * 3 * 2 * 1 = 5! = 120 different ways exist for the five teams to be ranked.

(If this does not make sense, please see the combinations page, which contains a much more thorough explanation.)

## Simplifying Factorials

In dealing with factorials, expressions can often become quite complex. On the surface, it can look difficult to perform the computations necessary to arrive at the final answer, especially if you do not have a calculator. Consequently, understanding how to simplify factorials is extremely important.

Another Example: