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15B Weekly Quizzes
Fall 2000
Q1. A family has 4 girls and 3 boys.
(a) How many ways can they sit in a row?
Ans:
7!.
(b) How many ways can they sit in a row if boys and girls must alternate?
Ans:
4!
×
3! since the only possible seating pattern in GBGBGBG so we seat
the boys (4!) and seat the girls (3!).
Q2. How many ways can
t
teams each of size
s
be made from
st
people? The teams have
no names or other distinguishing features. Three versions were given depending on
student ID number:
s
=2
,t
=4;
s
=3
=3;
s
=4
.
Ans:
One could label the teams (count ordered teams). This will be
t
! times the num
ber of teams since there are
t
! ways to assign
t
labels to
t
diﬀerent sets of people. The
number of labeled teams is
(
st
s,s,.
..,s
)
and so the answer is
(
st
)!
t
!(
s
!)
t
. (One can do this with
repeated binomial coeﬃcients instead of multinomial coeﬃcients:
(
st
s
)(
s
(
t

1)
s
)
···
.)
Q3. Let
A
,
B
and
C
⊆
B
be sets. We make
B
A
into a probability space by selecting
functions from
A
to
B
uniformly at random.
(a) What is the probability that a random
f
is an injection?
Ans:
Since there are altogher
b
a
functions, each has probability 1
/b
a
. An injec
tion is an
a
list without repetition from
B
, so there are (
b
)
a
of them. Thus the
probability is (
b
)
a
/b
a
.
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 Fall '01
 Bender
 Math

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