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Math 455 Exam 1, University of Massachusetts, Spring 2005, Exams of Discrete Structures and Graph Theory

University of Massachusetts - AmherstDiscrete Structures and Graph Theory

An old exam paper from the math 455 course offered at the university of massachusetts in spring 2005. The exam consists of 9 questions covering various topics in mathematics, including combinations, permutations, probability, and number theory. Students were not allowed to use calculators, books, or notes during the exam, and each answer had to be justified. The exam is divided into six pages.

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

koofers-user-a7r
koofers-user-a7r 🇺🇸

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DEPARTMENT OF MATHEMATICS AND STATISTICS
UNIVERSITY OF MASSACHUSETTS
MATH 455 March 3, 2005
EXAM 1
Your Name:
Your Section:
This exam paper consists of 9 questions. It has 6 pages. You do not need a numerical
answer unless you are asked for it. It is okay to use ³a
b´,P(n, r), n! or abin your
answers. Each answer must be justified. No calculators, books or notes are allowed!
1. (10)
2. (10)
3. (10)
4. (10)
5. (10)
6. (10)
7. (10)
8. (20)
9 a. (10)
9 b. (10) (bonus)
TOTAL (110)
pf3
pf4
pf5

Partial preview of the text

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DEPARTMENT OF MATHEMATICS AND STATISTICS

UNIVERSITY OF MASSACHUSETTS

MATH 455 March 3, 2005 EXAM 1

Your Name:

Your Section:

This exam paper consists of 9 questions. It has 6 pages. You do not need a numerical answer unless you are asked for it. It is okay to use

(a b

) , P (n, r), n! or ab^ in your answers. Each answer must be justified. No calculators, books or notes are allowed!

9 a. (10)

9 b. (10) (bonus)

TOTAL (110)

  1. Give a numerical value for (^) ( 11 4

)

P (11, 2)

  1. How many solutions to x 1 + x 2 + x 3 + x 4 + x 5 = 50 are there in positive integers (i.e. each xi ≥ 1, i = 1, 2 , 3 , 4 , 5)?
  2. How many distinct arrangements of the letters in REVERSE start with an E and end with an R?
  1. What is the number of 4-letter words that either start or end with a vowel? (Note: “word” means a string letters, a vowel is one of the five letters {a, e, i, o, u}.)
  2. What is the probability that a number between 1 and 100 (inclusive) chosen at random will be divisible by either 2, 3 or 5?
  1. Prove by induction that for any n ≥ 1

1 1 · 3

(2n − 1)(2n + 1)

n 2 n + 1