Combinatorics Through Guided Discovery, 2004a

178 D. Hints to Selected Problems
the component that is a tree into a tree with one more vertex? To prove your
method works, use contradiction by assuming there is a spanning tree with
lower total cost.
119.a. If you have a spanning tree of G that contains e, is the graph that results from
that tree by contracting e still a tree?
122.c. If you decide to put it on a shelf that already has a book, you have two choices
of where to put it on that shelf.
122.e. Among all the places you could put books, on all the shelves, how many are
to the immediate left of some book? How many other places are there?
123. How can you make sure that each shelf gets at least one book before you start
the process described in Problem 122?
124. What is the relationship between the number of ways to distribute identical
books and the number of ways to distribute distinct books?
125. Look for a relationship between a multiset of shelves and a way of distributing
identical books to shelves
126. Note that ( n+k−1 )
k
= ( n+k−1 ).
n−1
So we have to figure out how choosing either k
elements or n − 1 elements out of n + k − 1 elements constitutes the choice of
a multiset. We really have no idea what set of n + k − 1 objects to use, so why
not use [n + k − 1]? If we choose n − 1 of these objects, there are k left over, the
same number as the number of elements of our multiset. Since our multiset
is supposed to be chosen from an n-element set, perhaps we should let the
n-element set be [n]. From our choice of n − 1 numbers, we have to decide
on the multiplicity of 1 through n. For example with n =4and k =6,we
have n + k − 1=9. Here, shown with underlines, is a selection of 3=n − 1
elements from [9]: 1, 2, 3, 45, 6, 7, 8, 9. How do the underlined elements give
us a multiset of size 6 chosen from an [4]-element set? In this case, 1 has
multiplicity 2, 2 has multiplicity 1, 3 has multiplicity 2, and 4 has multiplicity
1.
127. A solution to the equations assigns a nonnegative number to each of 1, 2,...,m
so that the nonnegative numbers add to r. Does such an assignment have a
combinatorial meaning?
128. Can you think of some way of guaranteeing that each recipient gets m objects
(assuming k ≥ mn) right at the beginning of the process of passing the objects
out?
129. We already know how to place k distinct books onto n distinct shelves so
that each shelf gets at least one. Suppose we replace the distinct books with