Math 215 - Discrete Mathematics

Most of the course deals with developing and applying techniques of counting.
 Here are some questions that can be answered by those techniques:

(1) The manager of a bookstore wants to arrange some books on a shelf.  No
two of the books look alike.  There are 6 red physics books, 3 blue physics
books, 5 green physics books, 4 red art books, 3 blue art books, and 2 green
art books.  In how many ways can these books be arranged so that all books of
the same color are grouped together and within each color all books on the
same subject are grouped together?

(2) A "bridge hand" is a set of 13 cards chosen from a standard deck. How
many bridge hands include cards from exactly 2 suits?

(3) How many ways are there to fill a box with a dozen doughnuts chosen from
five varieties with the requirement that at least one dougnut of each variety
is picked?

(4) Given 5 pairs of gloves and 5 people, how many ways are there for each of
the people to choose 2 gloves with no one getting a matching pair?

(5) Consider a hotel containing 8 rooms, and let r be a positive integer. 
Determine the number of ways in which r people can be placed into these rooms
so that Room 1 contains at least 1 person, Rooms 2 and 5 each contain an odd
number of people, and Room 8 contains at least 1 person.

(6) A Girl Scout troop with 7 members is selling boxes of cookies. Six of the
girls are each given 1 box to sell.  The seventh girl plans to sell cookies
to 2 families, and she knows that one of those families will buy 2 boxes or
none at all, while the other family will buy 5 boxes or none at all.  The
leader of the troop will prepare a ledger listing the first 6 girls
alphabetically and the seventh girl last, and stating the number of boxes
sold by each girl.  If 10 boxes are sold, how many ledgers are possible?

The course also includes a brief introduction to graph theory, which is the
mathematical theory of networks.  Some of the counting techniques developed
in the earlier part of the course are applied in the graph theory.