DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

◦81. Use the General Product Principle to prove the number of functions
from an m-element set to a n-element set is m n . A common notation
for the set of all functions from a set M to a set N is N M .
+ 82. Now suppose you are thinking about the set S of functions f from [m]
to [n]. (For example, the set of functions from the three possible places
for scoops in an ice-cream cone to 12 flavors of ice cream.) Suppose
f(1) can be chosen in k1 ways. (In the ice cream problem, k1 = 12
holds because there were 12 ways to choose the first scoop.) Suppose
that for each choice of f(1) there are k2 choices for f(2). (For the ice
cream cones, k2 = 12 when the second flavor could be the same as the
first, but k2 = 11 when the flavors had to be different.) In general,
suppose that for each choice of f(1), f(2), . . . , f(i − 1), there are ki
choices forf(i). What has been assumed so far about the functions in
S may be summarized as:
• There are k1 choices for f(1).
• For each choice of f(1), f(2), . . . , f(i − 1), there are ki choices
for f(i).
How many functions are in the set S? Is there any practical difference
between the result of this problem and the General Product Principle?
The point of Problem 82 is that originally the statement the General Product
Principle was somewhat informal. To be more mathematically precise it is
a statement about counting sets of functions.
83. This problem revisits the question: How many subsets does a set S
with n elements have? (Compare with Problems 26 and 57.)
(a) For the specific case of n = 3, describe a sequence of three decisions
which could be made to yield subsets of [3]. Apply the General
Product Principle to find the number of subsets of [3]. Re-work
this from a function point of view.
(b) Use the functional interpretation of the General Product Principle
to prove that a set with n elements has 2 n subsets.
•84. In how many ways can you pass out k distinct pieces of fruit to n
children (with no restriction on how many pieces of fruit a child may
get)?
◦85. Assuming k ≤ n, in how many ways can you pass out k distinct pieces
of fruit to n children if each child may get at most one? What is the
number if k > n? Assume for both questions that you pass out all the
fruit. Note that each of these is a list of k distinct things chosen from
a set S (of children).
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