182
Solutions to Review Exercises
6. Let 8 be the set of cars with stereo systems,
A
the set of cars with air conditioners and
R
the set with
sun roofs. We are given that
181
=
3D, IAI
=
3D, IRI
=
40, 1(8
n
A)
U
(8 n
R)
U
(A
n
R)
I
=
30
and
18
nAn
RI
=
10.
By the Principle of InclusionExclusion,
30 = 1(8
n
A)
U (8
n
R)
U
(A
n
R)I
=
18 n
AI
+
18 n
RI
+
IA
n
RI
1(8 n
A)
n (8 n
R)I
1(8 n
A)
n
(A
n
R)I
1(8 n
R)
n
(A
n
R)I
+
1(8 n
A)
n (8 n
R)
n
(A
n
R)I
=
18nAI
+ 18nRI +
IAnRI218nAnRI·
Thus,
18 n
AI
+
18 n
RI
+
IA
n
RI
= 30
+
2(10)
= 50. It
follows that the number of car with at least
one feature is
18 U
A
U
RI
= 181
+
IAI
+
IRI  18 n
AI

18 n
RI

IA
n
RI
+
18
nAn
RI
=
30
+
30
+
40 
50
+
10
=
60.
Since the number of cars with exactly one feature is the number with at least one feature less the
number with at least two, the number with exactly one is
60 
30 = 30.
7.
If
n
objects are put into m boxes and
n
> m, then some box must contain at least
r
{;,
1
objects.
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 Summer '10
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 Graph Theory, Natural number, Finite set, 1 3, 1 1 3

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