# 1. Factorial Notation

For the following sections on counting, we need a simple way of writing the product of all the positive whole numbers up to a given number. We use **factorial notation** for this.

## Definition of *n*!

** n factorial** is defined as the product of all the integers from 1 to

*n*(the order of multiplying does not matter) .

We write "*n* factorial" with an exclamation mark as follows: `n!`

n! = (n)(n− 1)(n− 2)...(3)(2)(1)

### Examples

a) 5! = 5 × 4 × 3 × 2 × 1 = 120

b) 10! = 10 × 9 × 8 ×... × 3 × 2 × 1 = 3,628,800

c) 0! = 1 (this is a convention)

d) 2! = 2

### Exercise

Find the value of: `(10!)/(5!)`

Answer

We write it out in full and cancel the portions in brackets, as follows:

`(10!)/(5!)=(10xx9xx8xx7xx6xx(5xx4xx3xx2xx1))/((5xx4xx3xx2xx1)) `

`=10xx9xx8xx7xx6 `

`=30240`

NOTE: We conclude from this answer and the answer for (d) above that we cannot simply cancel a fraction containing factorials. That is:

`(10!)/(5!)!=2!`

We use factorial notation throughout this chapter, starting in the Permutations section.