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.
