Cambridge Additional Mathematics

262 Counting and the binomial expansion (Chapter 10)

EXERCISE 10C.2

1 Use the formula

¡n

r

¢

=

n!

r!(n¡r)!

to evaluate:

a

¡ 3

1

¢

b

¡ 4

2

¢

c

¡ 7

3

¢

d

¡ 10

4

¢

Check your answers using technology.

2aUse the formula

¡n

r

¢

= n!

r!(n¡r)!

to evaluate:

i

¡ 8

2

¢

ii

¡ 8

6

¢

b Show that

¡n

r

¢

=

¡ n

n¡r

¢

for all n 2 Z+, r=0, 1 , 2 , ....,n.

3 Findkif

¡ 9

k

¢

=4

¡ 7

k¡ 1

¢

Apermutationof a group of symbols isany arrangementof those symbols in a definiteorder.

For example, BAC is a permutation on the symbols A, B, and C in which all three of them are used.

We say the symbols are “taken 3 at a time”.

The set of all the different permutations on the symbols A, B, and C taken 3 at a time, is

fABC, ACB, BAC, BCA, CAB, CBAg.

Example 8 Self Tutor

List the set of all permutations on the symbols P, Q, and R taken:

a 1 at a time b 2 at a time c 3 at a time.

a fP, Q, Rg b fPQ, QP, RP,

PR, QR, RQg

c fPQR, PRQ, QPR,

QRP, RPQ, RQPg

Example 9 Self Tutor

List all permutations on the symbols W, X, Y, and Z taken 4 at a time.

WXYZ

XWYZ

YWXZ

ZWXY

WXZY

XWZY

YWZX

ZWYX

WYXZ

XYWZ

YXWZ

ZXWY

WYZX

XYZW

YXZW

ZXYW

WZXY

XZYW

YZWX

ZYWX

WZYX

XZWY

YZXW

ZYXW

There are 24 of them.

For large numbers of symbols, listing the complete set of permutations is absurd. However, we can still

count them by considering the number of options we have for filling each position.

D Permutations

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_10\262CamAdd_10.cdr Monday, 23 December 2013 4:29:38 PM BRIAN