Homework 4, Math 245, Fall 2010
Due Wednesday, October 20 in class.
The solution of each exercise should be at most one page long. If you can, try to write
your solutions in LaTex. Each question is worth 2 points.
1. A club of 8 men and 10 women has to select a committee of 4 people with at least one
man and one woman. In how many ways can this be done ? If among the four committee
members, one is the chairperson, one is convener, one is secretary and one is treasurer,
in how many ways can the four members of the committee be chosen ?
Proof.
To count the number of committees of 4 people containing at least one man and
one woman, one may split the committees according to how many men and women they
contain:
(a) 1 man, 3 women: there are
(
8
1
)
×
(
10
3
)
such committees.
(b) 2 men, 2 women: there are
(
8
2
)
×
(
10
2
)
such committees.
(c) 3 men, 1 woman: there are
(
8
3
)
×
(
10
1
)
such committees.
Thus, there are
8
1
10
3
+
8
2
10
2
+
8
3
10
1
= 8
×
120 + 28
×
45 + 56
×
10 = 2780
ways to form such committee.
Another solution for this question is to notice that the number of committees we are
trying to find equals the total number of committees formed from 8 men and 10 women
minus the number of committees formed by 4 men and the number of committees formed
by 4 women. This number is
10 + 8
4

10
4

8
4
= 3060

210

70 = 2780
.
To count the number of ways to form a committee where the four positions of the
committee matter, one has to count the number of ways to assign the four people of
each of the 2780 committees found earlier to the 4 jobs. This number is 4
·
3
·
2 = 24
and the total number of committees will be 2780
·
24 = 66720.
2. What is the coefficient of
x
20
in the binomial expansion of
(
2
x
3

3
x
)
8
? What is the
coefficient of
x
12
in (1 +
x
+
x

1
)(1 +
x
)
26
?
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Proof.
We use the binomial formula which states
(
a
+
b
)
n
=
n
X
k
=0
n
k
a
k
b
n

k
.
Taking
a
= 2
x
3
, b
=

3
x
and
n
= 8, we get
2
x
3

3
x
8
=
8
X
k
=0
8
k
(2
x
3
)
k

3
x
8

k
=
8
X
k
=0
8
k
2
k
(

3)
8

k
x
3
k

(8

k
)
=
8
X
k
=0
8
k
2
k
(

3)
8

k
x
4
k

8
.
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 Fall '10
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