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A P P E N D I X
A
E
Counting methods and
the binomial theorem
A1
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Counting methods
The addition rule
In general, to choose among alternatives simply add up the available number for each
alternative.
Example 1
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At the library Alan is having trouble deciding which book to borrow. He has a choice between
three mystery novels, three biographies or two science fiction novels. How many choices of
book does he have?
Solution
Since he is choosing between alternatives (mystery novels or biographies or science
fiction), he has a total of 3 + 3 + 2 = 8 choices.
The multiplication rule
When sequential choices are involved, the total number of possibilities is found by multiplying
the number of options at each successive stage.
Example 2
Sandi has six choices of windcheaters or jackets, and seven choices of jeans or skirts. How
many choices does she have for a complete outfit?
Solution
Since Sandi will wear both a windcheater or jacket and jeans or a skirt, we cannot
consider these to be alternative choices. We could draw a tree diagram to list the
possibilities, but this would be arduous. Using the multiplication rule, however, we
can quickly determine the number of choices as 6 × 7 = 42.
695
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696
Essential Mathematical Methods 3 & 4 CAS
Permutation or arrangements
The number of permutations of n objects in groups of size r is denoted n Pr and:
n
n!
(n − r ) !
= n × (n − 1) × (n − 2) × · · · × (n − r + 1)
Pr =
E
Example 3
How many different four-figure numbers can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 if
each digit may be used only once?
Solution
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Since we are arranging nine objects in groups of four, the answer is:
9!
5!
=9×8×7×6
= 3024
P4 =
9
Combinations or selections
In general, the number of combinations of n objects in groups of size r is:
n
Pr
r!
n × (n − 1) × (n − 2) × · · · × (n − r + 1)
=
r!
n!
=
r !(n − r )!
n
A commonly used alternative notation for n Cr is
.
r
Cr =
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n
Example 4
Four flavours of ice cream are available at the school canteen: vanilla, chocolate, strawberry
and caramel. How many different double cone selections are possible if two different flavours
must be used?
Solution
The number of combinations of four ice creams in groups of size two is:
4
4!
2!2!
4×3
=
2×1
=6
C2 =
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Appendix A — Counting methods and the binomial theorem
697
Example 5
A team of three boys and three girls is to be chosen from a group of eight boys and five girls.
How many different teams are possible?
Solution
We can choose three boys from eight in 8 C3 ways, and three girls from five in 5 C3
ways. Thus the total number of possible teams is:
C3 × 5C3 = 56 × 10
= 560
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Exercise A1
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1 A student needs to select a two unit study for her course, one unit in each semester. In
semester 1 she must choose one of two mathematics units, three language units and four
science units. In semester 2 she has a choice of two history units, three geography units
and two art units. How many choices does she have for the complete course?
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2 In order to travel from Melbourne to Brisbane, Dominic is given the following choices.
He can fly directly from Melbourne to Brisbane on one of three airlines, or he can fly
from Melbourne to Sydney on one of four airlines and then travel from Sydney to
Brisbane with one of five bus lines, or he can go on one of three bus lines directly from
Melbourne to Brisbane. In how many ways could he travel from Melbourne to Brisbane?
3 If there are eight swimmers in the final of the 1500 m freestyle event, in how many ways
can the first three places be filled?
4 In how many ways can the letters of the word TROUBLE be arranged:
a if they are all used?
b in groups of three?
5 In how many ways can the letters of the word PANIC be arranged:
a if they are all used?
b in groups of four?
6 A student has the choice of three mathematics subjects and four science subjects. In how
many ways can she choose to study one mathematics and two science subjects?
7 A survey is to be conducted, and eight people are to be chosen from a group of 30.
a How many different groups of eight people could be chosen?
b If the group contains 10 men and 20 women, how many groups of eight people
containing exactly two men are possible?
8 From a standard 52-card deck, how many seven-card hands have exactly five spades and
two hearts?
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698
Essential Mathematical Methods 3 & 4 CAS
9 In how many ways can a committee of five be selected from eight women and four men:
a without restriction?
b if there must be three women on the committee?
10 Six females and five males are interviewed for five positions. If all are found to be
acceptable for any position, in how many ways could the following combinations be
selected?
A2
b four females and one male
d five people regardless of sex
Summation notation
Suppose m and n are integers with m < n. Then:
n
ai = am + am+1 + · · · + an
i=m
E
a three females and two males
c five females
e at least four females
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This notation, which is called summation notation or sigma notation, is very convenient
for concisely representing sums. These sums will arise throughout the course. The notation
uses the symbol , the upper case form of the Greek letter sigma.
n
ai is read ‘the sum of the numbers ai from i equals m to i equals n’.
The symbol
i=m
The expression am + am+1 + · · · + an is called the expanded form of
n
ai .
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i=m
Example 1
Write
5
2i in expanded form and evaluate.
i=1
Solution
5
i=1
2i = 21 + 22 + 23 + 24 + 25
= 2 + 4 + 8 + 16 + 32
= 62
Example 2
Write 12 + 22 + 32 + · · · + 302 using summation notation.
Solution
12 + 22 + 32 + · · · 302 =
30
k2
k=1
Example 3
Write x1 + x2 + x3 + · · · + x11 using summation notation.
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Appendix A — Counting methods and the binomial theorem
699
Solution
x1 + x2 + x3 + · · · x11 =
11
xi
i=1
Exercise A2
E
1 Write each of the following in expanded form and evaluate:
5
5
5
4
1
(−1)i i
c
b
k3
d
i
i3
a
5 i=1
i=1
k=1
i=1
4
4
6
6
1
(k − 1)2
f
i
e
h
i2
(i − 2)2
g
k=1
3 i=1
i=1
i=1
2 Write each of the following using summation notation:
a 1 + 2 + 3 + ··· + n
x1 + x2 + x3 + · · · + x10
c
10
1 1 1 1
e 1+ + + +
2 3 4 5
b x 1 + x2 + x3 + · · · + x11
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d 14 + 24 + 34 + · · · + n 4 + (n + 1)4
3 Write each of the following in expanded form:
n
5
6
xi
a
b
x i · 25−i
c
x i · 2i · 36−i
i=1
i=0
i=0
d
4
(x − xi )i
i=0
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4 Write each of the following using summation notation:
a x 5 + 3x 4 + 9x 3 + 27x 2 + 81x + 243
c 4x 2 + 2x + 1
A3
b x 5 − 3x 4 + 9x 3 − 27x 2 + 81x − 243
d 8x 3 + 12x 2 + 18x + 27
The binomial theorem
Consider the expansion of binomial powers shown below:
(x
(x
(x
(x
(x
(x
+ b)0
+ b)1
+ b)2
+ b)3
+ b)4
+ b)5
=1
= 1x + 1b
= 1x 2 + 2xb + 1b2
= 1x 3 + 3x 2 b + 3xb2 + 1b3
= 1x 4 + 4x 3 b + 6x 2 b2 + 4xb3 + 1b4
= 1x 5 + 5x 4 b + 10x 3 b2 + 10x 2 b3 + 5xb4 + 1b5
The coefficients can be arranged as shown:
1
1
1
1
1
1
3
4
5
1
2
1
3
6
10
1
4
10
1
5
1
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700
Essential Mathematical Methods 3 & 4 CAS
This array is known as Pascal’s triangle. It was observed in Essential Mathematical Methods
1 & 2 CAS that the array can be constructed from combinations:
4
0
3
0
2
0
4
1
1
0
3
1
0
0
2
1
4
2
Row
0
1
1
3
2
1
2
2
2
3
3
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4
3
3
4
4
4
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n!
n
Remember:
=
(n − r ) !r !
r
The expansion of (x + b)8 can be written by utilising this observation:
8
8
8
8
8
(x + b)8 =
x8 +
x 7b +
x 6 b2 +
x 5 b3 +
x 4 b4
0
1
2
3
4
8
8
8
8
+
x 3 b5 +
x 2 b6 +
xb7 +
b8
5
6
7
8
In summation notation:
8 8−k k
(x + b)8 =
x b
k
k=0
8
In general:
n
(x)n−k bk
(x + b)n =
k
k=0
n
n
(ax)n−k bk
(ax + b)n =
k
k=0
n
and
n
(ax)n
The first term of the expansion of (ax + b) is
0
n
(ax)n−1 b
The second term is
1
n
(ax)n−r br
The (r + 1) th term is
r
It is convention that the binomial expansion of (ax + b)n is written in decreasing exponents
of x.
n
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Appendix A — Counting methods and the binomial theorem
701
Example 4
Find the coefficient of x20 in the expansion of (x + 2)30 .
Solution
30
The (r + 1)th term is
x 30−r 2r
r
When 30 − r = 20, r = 10
30
20
∴ the term with x is
210 x 20
20
20
The coefficient of x is
Example 5
Expand (2x + 3)5 .
Solution
30
20
210
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5
(2x)5−k 3k
(2x + 3)5 =
k
k=0 5
5
5
5
(2x)5 +
(2x)4 · 3 +
(2x)3 · 32 +
(2x)2 · 33
=
0
1
2
3
5
(2x) · 34 + 35
+
4
= 32x 5 + 240x 4 + 720x 3 + 1080x 2 + 810x + 243
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5
Example 6
Find the eighth term in the expansion of (2x – 4)10 .
Solution
The (r + 1)th term is
∴ the 8th term is
10
r
10
7
(2x)10−r · (−4)r
(2x)3 · (−4)7
= −15 728 640x 3
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702
Essential Mathematical Methods 3 & 4 CAS
Exercise A3
1 Expand each of the following using the binomial theorem:
a (x + 6)6
e (2x − 6)6
b (2x + 1)5
f (2x − 3)4
c (2x − 1)5
g (x − 2)6
d (2x + 3)6
h (x + 1)10
2 Find the eighth term in the expansion of:
c (1 − 2x)10
f (2x − b)12
E
a (2x − 1)10
b (2x + 1)10
12
d (3x + 1)
e (x + 3)12
(Descending powers of x are assumed.)
1 9
assuming descending powers of x.
3 Find the third term in the expansion of 2 − x
3
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4 Find the sixth term in the expansion of (3x − 1)11 assuming descending powers of x.
5 Expand (1 − x)11 .
6 Find the coefficient of x3 in the expansion of each of the following:
a (x + 2)5
d (4x − 3)7
b (2x − 1)6
e (3x + 4)4
c (1 − 2x)5
f (3x − 2)5
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7 Find the coefficient of x10 in the expansion of (2x − 3)14 .
8 Find the coefficient of x5 in the expansion of (4 − 2x)6 .
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