MACM 201 Midterm 1 Exam - Mathematics, Exams of Discrete Mathematics

A midterm exam for the macm 201 course at simon fraser university, department of mathematics, held in spring 2008. The exam covers various mathematical concepts, including set theory, combinatorics, and recurrence relations. Students are required to answer questions related to the principle of inclusion and exclusion, arranging letters in a word, solving equations with integer solutions, and arranging cars on a parking lot.

Typology: Exams

2012/2013

Uploaded on 02/21/2013

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SIMON FRASER UNIVERSITY
DEPARTMENT OF MATHEMATICS
Midterm 1
MACM 201 Spring 2008
Instructor: Robert ˇ
amal
February 6, 2008, 12:30 13:20
Name: (please print)
family name given name
SFU ID:
student number SFU-email
Signature:
Instructions:
1. Do not open this booklet until told to do so.
2. Write your name above in block letters. Write your
SFU student number and email ID on the line pro-
vided for it.
3. Write your answer in the space provided below the
question. If additional space is needed then use the
back of the previous page. Your final answer should
be simplified as far as is reasonable.
4. Make the method you are using clear in every case
unless it is explicitly stated that no explanation is
needed.
5. This exam has 5 questions on 6 pages (not includ-
ing this cover page). Once the exam begins please
check to make sure your exam is complete.
6. No calculators, books, papers, or electronic devices
shall be within the reach of a student during the
examination. The only exception is your formula
sheet—a one-sided sheet of paper.
7. During the examination, communicating with,
or deliberately exposing written papers to the
view of, other examinees is forbidden.
Question Maximum Score
1 10
2 10
3 10
4 10
5 10
Total 50
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SIMON FRASER UNIVERSITY

DEPARTMENT OF MATHEMATICS

Midterm 1

MACM 201 Spring 2008 Instructor: Robert ˇS´amal February 6, 2008, 12:30 – 13:

Name: (please print) family name given name

SFU ID: student number SFU-email

Signature:

Instructions:

  1. Do not open this booklet until told to do so.
  2. Write your name above in block letters. Write your SFU student number and email ID on the line pro- vided for it.
  3. Write your answer in the space provided below the question. If additional space is needed then use the back of the previous page. Your final answer should be simplified as far as is reasonable.
  4. Make the method you are using clear in every case unless it is explicitly stated that no explanation is needed.
  5. This exam has 5 questions on 6 pages (not includ- ing this cover page). Once the exam begins please check to make sure your exam is complete.
  6. No calculators, books, papers, or electronic devices shall be within the reach of a student during the examination. The only exception is your formula sheet—a one-sided sheet of paper.
  7. During the examination, communicating with, or deliberately exposing written papers to the view of, other examinees is forbidden.

Question Maximum Score

Total 50

1. Let S be a (finite) set, and c 1 , c 2 , c 3 conditions on the elements of S.

[1] (a) State the Principle of Inclusion and Exclusion, and explain the used notation. Don’t use ‘... ’, nor ‘

[2] (b) Decide whether the following formulas are true or false, and if false, correct them!

  1. N( c 2 ) = N − N( c 2 )
  2. N( c 2 c 3 ) = N( c 1 ) − N( c 2 c 3 )

[2] (c) Which elements of S are counted by N − N( c 1 c 2 c 3 ) − N( c 1 c 2 c 3 )?

2. In how many ways can five A’s, five B’s and five C’s be arranged to make a 15-letter

word, if we require that there is

[7] (a) no consecutive fivetuple of the same letter?

[3] (b) exactly one consecutive fivetuple of the same letter?

3. How many solutions there are to the equation

x 1 + x 2 + x 3 + x 4 = 10

such that x 1 ,... , x 4 are integers and

[3] (a) x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 , x 4 ≥ 0?

[7] (b) 0 ≤ x 1 ≤ 4 , 0 ≤ x 2 ≤ 4 , 0 ≤ x 3 ≤ 4 , 0 ≤ x 4 ≤ 4?

[8] 5. (a) Solve the following recurrence relation.

an+2 + 2an+1 − 8 an = 0 (n ≥ 0), a 0 = 0, a 1 = 1.

[2] (b) Check the answer for n = 2.