Counted - Discrete Mathematics - Exam, Exams of Discrete Mathematics

This is the Exam of Discrete Mathematics which includes Recurrence Relation, Space Is Available, Answer, Number of Ways, Sum of Odd Integers, By Hand, Solution, Various Walks, Provided etc. Key important points are: Counted, Convolution, Sequence, Generating Functions F, Coefficient, Recurrence Relation, Sequence, Generating Function, Nonnegative Integers, Number of Solutions

Typology: Exams

2012/2013

Uploaded on 02/21/2013

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SIMON FRASER UNIVERSITY
DEPARTMENT OF MATHEMATICS
Midterm 2
MACM 201 Summer 2007
Instructor: Robert ˇ
amal
July 4, 2007, 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 5 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 8
2 12
3 10
4 10
5 10
Total 50
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pf4
pf5

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SIMON FRASER UNIVERSITY

DEPARTMENT OF MATHEMATICS

Midterm 2

MACM 201 Summer 2007 Instructor: Robert ˇS´amal July 4, 2007, 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 5 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

[2] 1. (a) Evaluate

4 2

[2] (b) Explain why p(4) = 5. (That is, list all objects that are counted by p(4).)

[4] (c) Let an = 2 and bn = n for all n ≥ 0. Suppose that the sequence c 0 , c 1 , c 2 ,... is the convolution of a 0 , a 1 , a 2 ,... and b 0 , b 1 , b 2 ,.... What is cn?

[10] 3. Let an be given by the following recurrence relation

an − 2 an− 1 = 3n^ (n ≥ 1), a 0 = 3.

Find the generating function for the sequence a 0 , a 1 , a 2 ,....

4. This question concerns finding the number of solutions to the equation

a + b + c = 100 ,

where a, b, c are nonnegative integers such that

  • a is even,
  • b is odd, and
  • c ≥ 5.

[3] (a) Express the answer as a coefficient of some power of x in an appropriate generating function f (x).

[4] (b) Express f (x) as a polynomial (1−x (^2) ) 3.

[3] (c) Use the binomial theorem to get the final answer.