Combinatorics Through Guided Discovery, 2004a

170 D. Hints to Selected Problems
15. In how many ways may you assign the men to their rows? The women? Once
a woman and a man have a row to share, in how many ways may they choose
their seats?
18. Try applying the product principle in the case n =2and n =3. How might
you apply it in general?
19. Ask yourself if either the sum principle or product principle applies.
Additional Hint: Remember that zero is a number.
20. Do you see an analogy between this problem and any of the previous problems?
26.a. For each part of this problem, think about how many arrows are allowed to
enter a vertex representing a member of Y.
28. The problem is asking you to describe a one-to-one function from the set of
binary representations of numbers between 0 and 2 n −1 onto the set of subsets
of the set [n]. Write down these two sets for n =2. They should both have
four elements. The set of binary representations should contain the string 00.
You could think of this as the instruction “take no ones and take no twos.”
In that context, what could you think of the string 11 as standing for? This
should help you describe a function. Of course now you have to figure out
how to show it is one-to-one and onto.
31. Starting with the row 1 8 28 56 70 56 28 8 1, put dots below it where the
elements of row 9 should be. Then put dots below that where the elements
of row 10 should be. Do the same for rows 11 and 12. Mark the dot where
row 12 should appear. Now mark the dots you need in row 11 to compute
the entry in column 3 of row 12. Now mark the dots you need in row 10 to
compute the marked entries in row 11. Do the same for rows 9 and 8. Now
you should be able to see what you need to do.
32.a. Begin by trying to figure out what the entries just above the diagonal of the
rectangle are. After that, what other entries can you figure out?
32.b. See if you can figure out what the entries in column −1 have to be.
32.c. What does the sum of two consecutive values in row −1 have to be? Could
this sum depend on which two consecutive values we take? Is there some
value of row −1 that we could choose arbitrarily? Now what about row −2?
Can we make arbitrary choices there? If so, how many can we make, and is
their position arbitrary?
36. The first thing you need to decide is “What are the two sets whose elements
we are counting?” Then it will be easier to think of a bijection between these
two sets. It turns out that these two sets are sets of sets!