Page 33 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series
In general, n! is defined as

+
n! = n(n – 1)(n – 2) … 3 · 2 · 1, with n  Z ,
and 0! = 1.
Notice that
(n + 1)! = (n + 1)[n(n – 1)(n – 2) … 3 · 2 · 1]
= (n + 1) · n!

Example 29

Evaluate 10! .
2 8!
Solution: 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
8! 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
= 10 × 9
= 90
Alternative method:
10! = 10 × 9! By using (n + 1)! = (n + 1) × n!
8! 8!
= 10 × 9 × 8!
8!
= 90

Example 30

Evaluate 18! .
15! 4!
Solution: 18! = 18 × 17 × 16 × 15!
15! 4! 15! × 4!
= 18 × 17 × 16
4 × 3 × 2 × 1
= 204

Example 31

Rewrite each of the following products in factorial form.
(a) 45 × 44 × 43 × 42 × 41 (b) 10 × 9 × 8
5 × 4 × 3
Solution: (a) 45 × 44 × 43 × 42 × 41
= 45 × 44 × 43 × 42 × 41 × 40 × 39 × … × 3 × 2 × 1
40 × 39 × … × 3 × 2 × 1
= 45!
40!

(b) 10 × 9 × 8 = 10 × 9 × 8 × 7! × 2!
5 × 4 × 3 5 × 4 × 3 7! 2!
= (10 × 9 × 8 × 7!) × 2!
(5 × 4 × 3 × 2!) × 7!
= 10! 2!
5! 7!

120

02 STPM Math T T1.indd 120 3/28/18 4:21 PM