Combinatorics Through Guided Discovery, 2004a

Appendix D
Hints to Selected Problems
1. Answer the questions in Problem 2 for the case of five schools.
3. For each kind of bread, how many sandwiches are possible?
6. Try to solve the problem first with a two-scoop cone. (Look for an earlier
problem that is analogous.) Then, for each two scoop cone, in how many
ways can you put on a top scoop?
7.a. Ask yourself “how many choices do we have for f (1)?” Then ask how many
choices we have for f (2).
7.b. It may not be practical to write down rules for all the functions for this
problem. But you could ask yourself how many choices we have for f (1),
howmanywehavefor f (2) and how many we have for f (3).
7.c. If you are choosing a function f , how many choices do you have for f (a)?
Then how many choices do you have for f (b)?
8.a. You know how to figure out in how many ways they could make a list of
three flavors out of the twelve. But each set of three flavors can be listed in a
number of different ways. Try to figure out in how many ways a set of three
flavors can be listed, and then try to see how this helps you.
8.b. Try to break the problem up into cases you can solve by previous methods;
then figure out how to get the answer to the problem by using these answers
for the cases.
12.a. Suppose you have a list in alphabetical order of names of the members of the
club. In how many ways can you pair up the first person on the list? In how
many ways can you pair up the next person who isn’t already paired up?
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