5
UNIT 3
Questions 6 – 10
Attached to Runalong Fire Station there are seven firefighters (1, 2, 3, 4, 5, 6, 7). It is necessary to have three
firefighters at the station each night in case of emergency, and the Firefighters’ Union requires that each
firefighter works the same number of nights. Schedules I
–
IV were prepared for consideration.
I
II
III
IV
Sunday
{1,2,4}
{1,2,4}
{1,2,4}
{1,2,4}
Monday
{2,3,5}
{2,3,5}
{2,3,5}
{2,3,5}
Tuesday
{3,4,6}
{3,4,6}
{3,4,6}
{3,4,6}
Wednesday
{4,5,7}
{4,5,7}
{1,2,4}
{4,5,7}
Thursday
{1,2,4}
{5,6,1}
{5,6,1}
{5,6,1}
Friday
{2,3,5}
{6,7,2}
{6,7,2}
{7,1,2}
Saturday
{3,4,6}
{7,1,3}
{7,1,3}
{6,1,3}
6
Which one of the schedules meets the requirements of the Firefighters’ Union?
A
I
C
III
B
II
D
IV
Questions 7 – 10 refer to the following additional information:
I – IV are examples of schedules. A schedule is a set of
v
objects (in this case, firefighters) that have to be
arranged into
b
sets (in this case, one set for each day of the week) all of size
k
and such that each object occurs
the same number of times (
r
) in the schedule and only once in any set. For the firefighters’ schedules,
v
= 7,
b
= 7,
k
= 3, and
r
= 3.
7
If
v
= 3,
b
= 6,
k
= 1,
r
= 2, which one of the following completes the schedule {1}, {2}, {3}, {1}, {2}, . . .?
A
{1}
C
{3}
B
{2}
D
neither
A
, nor
B
, nor
C
8
The schedule {1,2}, {2,3}, {
x
,
y
} is a schedule for which
v
= 3,
b
= 3,
k
= 2,
r
= 2, if
A
x
= 1,
y
= 2 .
C
x
= 2,
y
= 2 .
B
x
= 1,
y
= 3 .
D
x
= 2,
y
= 3 .
9
The schedule
{1,2,3}
{4,5,6} {7,8,9} {1,4,7}
{2,5,8}
{3,6,9}
{1,5,9}
{2,6,7}
{3,4,8}
{1,6,8}
{2,4,9}
{
x
,
y
,
z
}
is a schedule for which
v
= 9,
b
= 12,
k
= 3,
r
= 4, if
A
x
= 1,
y
= 2,
z
= 4 .
C
x
= 2,
y
= 4,
z
= 6 .
B
x
= 1,
y
= 3,
z
= 5 .
D
x
= 3,
y
= 5,
z
= 7 .
10
Which one of the following equations is true for
all
schedules?
A
bk
=
rv
C
kv
= (
b
– 3) (
r
– 1)
B
br
=
kv
D
b
–
v
=
k
–
r