Combinatorics Through Guided Discovery, 2004a

1.2. Basic Counting Principles 17
In Table 1.2.8 we list all three-element permutations from the 5-element set
{a, b, c, d, e}. Each row consists of all 3-element permutations of some subset
of {a, b, c, d, e}. Because a given k-element subset can be listed as a k-element
permutation in k! ways, there are 3! = 6 permutations in each row. Because each
3-element permutation appears exactly once in the table, each row is a block of a
partition of the set of 3-element permutations of {a, b, c, d, e}. Each block has size
six. Each block consists of all 3-element permutations of some three-element subset
of {a, b, c, d, e}. Since there are ten rows, we see that there are ten 3-element subsets
of {a, b, c, d, e}. An alternate way to see this is to observe that we partitioned the set
of all 60 three-element permutations of {a, b, c, d, e} into some number q of blocks,
each of size six. Thus by the product principle, q · 6=60,soq =10.
• Problem 39. Rather than restricting ourselves to n =5and k =3, we can
partition the set of all k-element permutations of S up into blocks. We do so
by letting B K be the set (block) of all k-element permutations of K for each k-
element subset K of S. Thus as in our preceding example, each block consists
of all permutations of some subset K of our n-element set. For example,
the permutations of {a, b, c} are listed in the first row of Table 1.2.8. In fact
each row of that table is a block. The questions that follow are about the
corresponding partition of the set of k-element permutations of S, where S
and k are arbitrary.
(a) How many permutations are there in a block? (h)
(b) Since S has n elements, what does problem 20 tell you about the total
number of k-element permutations of S?
(c) Describe a bijection between the set of blocks of the partition and the
set of k-element subsets of S. (h)
(d) What formula does this give you for the number ( n )
k
of k-element
subsets of an n-element set? (h)
⇒
Problem 40. A basketball team has 12 players. However, only five players
play at any given time during a game.
(a) In how may ways may the coach choose the five players?
(b) To be more realistic, the five players playing a game normally consist
of two guards, two forwards, and one center. If there are five guards,
four forwards, and three centers on the team, in how many ways can
the coach choose two guards, two forwards, and one center? (h)
(c) What if one of the centers is equally skilled at playing forward? (h)