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37. Ask yourself “What is a problem like this doing in the middle of a bunch of
problems about counting subsets of a set? Is it related, or is it supposed to
gives us a break from sets?”
38. The problem suggests that you think about how to get a list from a seating
arrangement. Could every list of n distinct people come from a seating chart?
How many lists of n distinct people are there? How many lists could we get
from a given seating chart by taking different starting places?
Additional Hint: For a different way of doing the problem, suppose that you
have chosen one person, say the first one in a list of the people in alphabetical
order by name. Now seat that person. Does it matter where they sit? In ways
can you seat the remaining people? Does it matter where the second person
in alphabetical order sits?
39.a. A block consists of all permutations of some subset {a 1 , a 2 ,...,a k } of S. How
many permutations are there of the set {a 1 , a 2 ,...,a k }?
39.c. What sets are listed, and how many times is each one listed if you take one
list from each row of Table 1.2.8? How does this choice of lists give you the
bijection in this special case?
39.d. You can make good use of the product principle here.
40.b. The coach is making a sequence of decisions. Can you figure out how many
choices the coach has for each decision in the sequence?
40.c. As with any counting problem whose context does not suggest an approach,
it is useful to ask yourself if you could decompose the problem into simpler
parts by using either the sum or product principle.
43. How could we get a list of beads from a necklace?
Additional Hint: When we cut the necklace and string it out on a table, there
are 2n lists of beads we could get. Why is it 2n rather than n?
44.a. You might first choose the pairs of people. You might also choose to make a
list of all the people and then take them by twos from the list.
44.b. You might first choose ordered pairs of people, and have the first person in
each pair serve first. You might also choose to make a list of all the people
and then take them by twos from the list in order.
45. It might be helpful to just draw some pictures of the possible configurations.
There aren’t that many.
47. Note that we must walk at least ten blocks, so ten is the smallest number of
blocks possible. In how many of those ten blocks must we walk north?