Combinatorics - MATH 0345
Exam 1
October 13, 2006
Name:
Honor Code Pledge
Signature
Directions: Please complete all but 1 problem. There is a time limit of 2 hours.
1. Show that a magic square of order 3 must have a 5 in the middle position. Deduce
that there are exactly 8 magic squares of order 3.
2. Prove that in a simple graph with nvertices, with n > 1, that there exist at least
two vertices of the same degree.
3. Prove that r(
k+1
z }| {
3,3, . . . 3) ≤(k+ 1)(r(
k
z }| {
3,3, . . . 3) −1) + 2.
4. A football team of 11 players is to be selected from a set of 15 players, 5 of whom
can play only in the backfield, 8 of whom can play only on the line, and 2 of whom
can play either in the backfield or on the line. Assuming a football team has 7 men
on the line and 4 in the backfield, determine the number of football teams possible.
5. Twenty different books are to be put on five book shelves, each of which holds at
least twenty books.
•How many different arrangements are there if you only care about the number
of books on the shelves (and not which book is where)?
•How many different arrangements are there if you care about which books are
where but the order of the books on the shelves doesn’t matter?
•How many different arrangements are there if the order on the shelves does
matter?
6. What is the number of integral solutions of the equation:
1
