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