Cambridge Additional Mathematics

nn!is read “ factorial”.

P Q

Q

P

A

B

C

E

D

Counting and the binomial expansion (Chapter 10) 259

cd

2 Katie is going on a long journey to visit her family. She

lives in city A and is travelling to city E. Unfortunately

there are no direct trains. However, she has the choice

of several trains which stop in different cities along the

way. These are illustrated in the diagram.

How many different train journeys does Katie have to

choose from?

In problems involving counting, products of consecutive positive integers such as 8 £ 7 £ 6 and

6 £ 5 £ 4 £ 3 £ 2 £ 1 are common.

For convenience, we introducefactorial numbersto represent the products of consecutive positive integers.

For n> 1 , n! is the product of the firstnpositive integers.

n!=n(n¡1)(n¡2)(n¡3)::::£ 3 £ 2 £ 1

For example, the product 6 £ 5 £ 4 £ 3 £ 2 £ 1 can be written as6!.

Notice that 8 £ 7 £ 6 can be written using factorial numbers only as

8 £ 7 £6=

8 £ 7 £ 6 £ 5 £ 4 £ 3 £ 2 £ 1

5 £ 4 £ 3 £ 2 £ 1

=

8!

5!

An alternativerecursive definitionof factorial numbers is n!=n£(n¡1)! for n> 1

which can be extended to n!=n(n¡1)(n¡2)! and so on.

Using the factorial rule with n=1, we have 1! = 1£0!

Therefore, for completeness we define 0! = 1

Example 3 Self Tutor

Simplify: a 4! b

5!

3!

c

7!

4!£3!

a 4! = 4£ 3 £ 2 £1=24

b

5!

3!

=

5 £ 4 £ 3 £ 2 £ 1

3 £ 2 £ 1

=5£4=20

c

7!

4!£3!

=

7 £ 6 £ 5 £ 4 £ 3 £ 2 £ 1

4 £ 3 £ 2 £ 1 £ 3 £ 2 £ 1

=35

C Factorial notation

4037 Cambridge

cyan magenta yellow black Additional Mathematics

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\CAM4037\CamAdd_10\259CamAdd_10.cdr Friday, 4 April 2014 1:45:51 PM BRIAN