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MAA AMC
AMC 12/AHSME
1959 AMC 12/AHSME
50
50
Part of
1959 AMC 12/AHSME
Problems
(1)
Making committees
Source: 1959 AHSME Problem 50
8/14/2013
A club with
x
x
x
members is organized into four committees in accordance with these two rules:
(1) Each member belongs to two and only two committees
\text{(1)}\ \text{Each member belongs to two and only two committees}\qquad
(1)
Each member belongs to two and only two committees
(2) Each pair of committees has one and only one member in common
\text{(2)}\ \text{Each pair of committees has one and only one member in common}
(2)
Each pair of committees has one and only one member in common
Then
x
x
x
:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
cannont be determined
<span class='latex-bold'>(A)</span> \ \text{cannont be determined} \qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
cannont be determined
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
has a single value between 8 and 16
<span class='latex-bold'>(B)</span> \ \text{has a single value between 8 and 16} \qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
has a single value between 8 and 16
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
has two values between 8 and 16
<span class='latex-bold'>(C)</span> \ \text{has two values between 8 and 16} \qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
has two values between 8 and 16
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
has a single value between 4 and 8
<span class='latex-bold'>(D)</span> \ \text{has a single value between 4 and 8} \qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
has a single value between 4 and 8
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
has two values between 4 and 8
<span class='latex-bold'>(E)</span> \ \text{has two values between 4 and 8} \qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
has two values between 4 and 8
AMC