Combinatorics Through Guided Discovery, 2004a

1.2. Basic Counting Principles 7
+ Problem 13. Let us now return to Problem 7 and justify—or perhaps finish—our
answer to the question about the number of functions from a
three-element set to a 12-element set.
(a) How can you justify your answer in Problem 7 to the question “How
many functions are there from a three element set (say [3] = {1, 2, 3})
to a twelve element set (say [12])?”
(b) Based on the examples you’ve seen so far, make a conjecture about
how many functions there are from the set
[m] ={1, 2, 3,...,m}
to [n] ={1, 2, 3,...,n} and prove it.
(c) A common notation for the set of all functions from a set M to a set N
is N M . Why is this a good notation?
+ Problem 14. Now suppose we are thinking about a set S of functions f from
[m] to some set X. (For example, in Problem 6 we were thinking of the set of
functions from the three possible places for scoops in an ice-cream cone to
12 flavors of ice cream.) Suppose there are k 1 choices for f (1). (In Problem 6,
k 1 was 12, because there were 12 ways to choose the first scoop.) Suppose
that for each choice of f (1) there are k 2 choices for f (2). (For example, in
Problem 6 k 2 was 12 if the second flavor could be the same as the first, but
k 2 was 11 if the flavors had to be different.) In general, suppose that for each
choice of f (1), f (2),... f (i − 1), there are k i choices for f (i). (For example,
in Problem 6, if the flavors have to be different, then for each choice of f (1)
and f (2), there are 10 choices for f (3).)
What we have assumed so far about the functions in S may be summarized
as
• There are k 1 choices for f (1).
• For each choice of f (1), f (2),..., f (i − 1), there are k i 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 14 is that the general product principle can be stated
informally, as we did originally, or as a statement about counting sets of standard
concrete mathematical objects, namely functions.
⇒
Problem 15. A roller coaster car has n rows of seats, each of which has room
for two people. If n men and n women get into the car with a man and a
woman in each row, in how many ways may they choose their seats? (h)