Simplify
7!¡6!
6
by factorising.Counting and the binomial expansion (Chapter 10) 2615 Write as a product by factorising:
a 5! + 4! b 11!¡10! cd12!¡10!
e 9! + 8! + 7! f 7!¡6! + 8! g 12!¡ 2 £11! h 3 £9! + 5£8!Example 6 Self Tutor
6 Simplify by factorising:a 12!¡11!
11b 10! + 9!
11c 10!¡8!
89d 10!¡9!
9!e
6! + 5!¡4!
4!
f
n!+(n¡1)!
(n¡1)!
g
n!¡(n¡1)!
n¡ 1
h
(n+ 2)! + (n+ 1)!
n+3THE BINOMIAL COEFFICIENT
Thebinomial coefficientis defined by
¡n
r¢
=n(n¡1)(n¡2)::::(n¡r+ 2)(n¡r+1)
r(r¡1)(r¡2):::: 2 £ 1
| {z }
factor form=n!
r!(n¡r)!
| {z }
factorial formThe binomial coefficient is sometimes written nCr or Crn.Example 7 Self Tutor
Use the formula¡n
r¢
=
n!
r!(n¡r)!
to evaluate: a¡ 5
2¢
b¡ 11
7¢a¡ 5
2¢
=
5!
2!(5¡2)!=
5!
2!£3!=10b¡ 11
7¢
=
11!
7!(11¡7)!=
11!
7!£4!=
7920
24
= 3307!¡6!
6=
7 £6!¡6!
6=
6!(7¡1)
6
=6!11=
5 £ 4 £ 3 £ 2 £ 1
2 £ 1 £ 3 £ 2 £ 1
=
11 £ 10 £ 9 £ 8 £ 7 £ 6 £ 5 £ 4 £ 3 £ 2 £ 1
7 £ 6 £ 5 £ 4 £ 3 £ 2 £ 1 £ 4 £ 3 £ 2 £ 17! + 9!4037 Cambridge
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Y:\HAESE\CAM4037\CamAdd_10\261CamAdd_10.cdr Wednesday, 29 January 2014 9:04:43 AM BRIAN