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Combinatorics Exam 1 - MATH 0345, Exams of Mathematics

The october 13, 2006 exam for the combinatorics course, math 0345. The exam covers various topics including magic squares, simple graphs, combinations, and the binomial theorem. Questions include proving identities, using combinatorial reasoning, and expanding expressions using the binomial theorem.

Typology: Exams

2012/2013

Uploaded on 02/12/2013

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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
pf2

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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 n vertices, with n > 1, that there exist at least two vertices of the same degree.
  3. Prove that r(

k+ ︷ ︸︸ ︷ 3 , 3 ,... 3) ≤ (k + 1)(r(

k ︷ ︸︸ ︷ 3 , 3 ,... 3) − 1) + 2.

  1. 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.
  2. 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?
  3. What is the number of integral solutions of the equation:

z 1 + z 2 + z 3 = 11 if z 1 ≥ 2 and z 2 > z 3.

  1. Use combinatorial reasoning to prove the identity (in the form given).

( n k

n − 3 k

n − 1 k − 1

n − 2 k − 1

n − 3 k − 1

(Hint: Let S be a set with three distinguished elements a, b and x and count certain k-combinations of S.)

  1. Expand ( 12 z + 4x)^5 using the binomial theorem.
  2. There are two circular tables, each of which sit four. Given 7 people how many ways are there to make a seating arrangement for these 7 people.