MATH 55
Prof. Bernoff
Fall 2007
Harvey Mudd College
Homework 3 – Solutions
17.6 Prove:
n
k
=
k
+ 1
n

1
.
Solution:
There is a bijective argument that these two are equal using the idea of two arrangements
that are
duals
 that is two arrangements that can be made into each other by exchanging how objects
are labelled. The number of ways to arrange
k
bars and
n

1 stars is the same way as the number
of ways to arrange
k
stars and
n

1 bars. By the stars and bars argument,
((
n
k
))
counts the number
of ways to place
k
stars around
n

1 bars. Similarly,
((
k
+1
n

1
))
counts the number of ways to place
n

1 stars around
k
bars, and so they’re equal.
17.7 Let
n
k
denote the number of multisets of cardinality
k
we can form choosing the elements in
{
1
,
2
,
3
, ..., n
}
with the added condition that we must use each of these
n
elements at least once in the multiset.
(a) Evaluate from first principles,
n
n
.
Solution:
There is only one way to pick a set of
n
elements from
{
1
,
2
,
3
, ..., n
}
with each
number represented at least once.
(b) Prove:
n
k
=
(
n
k

n
)
.
Solution:
After choosing each of 1
, . . . , n
for a multiset of size
k
, we choose the remaining
k

n
elements as a multiset from
{
1
,
2
,
3
, ..., n
}
, so
n
k
=
((
n
k

n
))
.
17.8 Let
n, k
be positive integers. Prove:
n
k
=
n

1
0
+
n

1
1
+
n

1
2
· · ·
+
n

1
k
Solution:
There are
((
n
k
))
k
multisets of
{
1
,
2
,
3
, ..., n
}
, but we can count them all by conditioning
on how many times
n
appears in each multiset. The
k
multisets in which
n
appears
i
times are the
((
n

1
k

i
))
(
k

i
)multisets of
{
1
,
2
,
3
, ..., n

1
}
(which fill in the rest of each
k
multiset). Since
i
can
range from 0 to
k
, we have
((
n
k
))
=
∑
k
i
=0
((
n

1
k

i
))
.
17.9 Let
n, k
be positive integers. Prove:
n
k
=
1
k

1
+
2
k

1
+
3
k

1
· · ·
+
n
k

1
Solution:
We count the
k
multisets of
{
1
,
2
,
3
, ..., n
}
by conditioning on the largest element of each. If
i
is the largest element in a
k
multiset of
{
1
,
2
,
3
, ..., n
}
, there are
((
i
k

1
))
ways to choose the remaining
k

1 elements for the multiset. Since
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 Fall '07
 Bernoff
 Math, Numerical digit, Natural number, Playing card

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