Combinatorics Through Guided Discovery, 2004a

1.2. Basic Counting Principles 15
(b) We define a function f from the set of subsets of [n] ={1, 2,...,n} to
the set of functions from [n] to {0, 1} by f (S) =χ S . Explain why f is
a bijection.
(c) Why does the fact that f is a bijection prove that [n] has 2 n subsets?
a The symbol χ is the Greek letter chi that is pronounced Ki, with the i sounding like “eye.”
In Problems 18, Problem 28, and Problem 33 you gave three proofs of the
following theorem.
Theorem 1.2.7. The number of subsets of an n-element set is 2 n .
The proofs in Problem 28 and Problem 33 use essentially the same bijection,
but they interpret sequences of zeros and ones differently, and so end up being
different proofs. We will give yet another proof, using bijections similar to those
we used in proving the Pascal Equation, at the beginning of Chapter 2.
1.2.6 The quotient principle
• Problem 34. As we noted in Problem 29, the first question in Problem 8
asked us for the number of three-element subsets of a twelve-element set.
We were able to use the Pascal Equation to get a numerical answer to that
question. Had we had twenty or thirty flavors of ice cream to choose from,
using the Pascal Equation to get our answer would have entailed a good
bit more work. We have seen how the general product principle gives us
an answer to Problem 6. Thus we might think that the number of ways
to choose a three element set from 12 elements is the number of ways to
choose the first element times the number of ways to choose the second
element times the number of ways to choose the third element, which is
12 · 11 · 10 = 1320. However, our result in Problem 29 shows that this is
wrong.
(a) What is it that is different between the number of ways to stack ice
cream in a triple decker cone with three different flavors of ice cream
and the number of ways to simply choose three different flavors of ice
cream?
(b) In particular, how many different triple decker cones use the same
three flavors? (Of course any three distinct flavors could substitute for
vanilla, chocolate and strawberry without changing the answer.)
(c) Using your answer from part b, compute the number of ways to choose
three different flavors of ice cream (out of twelve flavors) from the
number of ways to choose a triple decker cone with three different
flavors (out of twelve flavors).