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Examination 3 for CMSC 203 Spring 2003 - Mathematics and Logic Problems, Exams of Discrete Structures and Graph Theory

English and Foreign Languages UniversityDiscrete Structures and Graph Theory

The third examination for the cmsc 203 course in spring 2003, focusing on mathematics and logic problems. It includes questions about license plates, combinations, orderings, seating arrangements, filling boxes, relations, and probabilities.

Typology: Exams

2012/2013

Uploaded on 04/27/2013

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CMSC 203 Spring 2003 Examination 3
1. (a) How many license plates can a state produce if the plates can contain 6 characters (from 26
letters and 10 digits) if they can only use one digit?
(b) How many ways can Mr. Paul choose 6 students from a class of 15 Boys and 12 Girls, if he
must choose at least 5 boys?
(c) How many orderings are there of the letters of the word STRAWBERRYALARMCLOCK ?
(d) How many ways can I seat 12 people around a circular table, if a certain pair of people cannot
sit next to one another?
(e) How many ways can I fill a box of 50 chocolates from 10 types if I must have at least 1 of each
type in the box?
2. Let R be the relation on Z given by R = {(a,b) | a,b ∈ Z and a ≡ b mod 5}.
(a) Prove the R is Reflexive. (b) Prove the R is Symmetric.
(c) Prove the R is Transitive. (d) Describe the partition of Z induced by R.
Let S be the relation on {1,2,3,4,5} given as
S = {(1,1),(1,3),(1,4),(2,1),(2,4),(3,1),(3,2),(3,3),(3,4),(3,5),(4,2),(4,5),(5,3),(5,4),(5,5)}
(e) Graph S-1. (f) Find MS, the Matrix of S. (g) Find MS o MS-1
(h) For the database whose entries form the following table:
Make Model Year Engine ID Vehicle ID Color
Ford Mustang 1972 A1222 FO13579 Black
Ford Fiesta 1989 C54322 FO24245 Yellow
Chevy Camaro 1991 754342AH CH172389 Black
Chevy Caprice 1989 442355CC CH156738 Yellow
Olds Cutlass 1992 ANDU33 OL64332 Blue
Olds Cutlass 1992 ANGH28 OL61998 White
Volvo P1800 1969 44325XX VO44526 White
Volvo 240 1986 53526PD VO64690 Black
Volvo 760 1992 578868R VO83529 Blue
find the Primary Keys and P3,6
3. (a) For a collection of 80 coins, if 53 are quarters, 15 are quarters from the 1990’s, and 24 are
coins from the 1990’s, what is the probability the a coin chosen at random is a quarter or is a coin
from the 1990’s?
(b) What is the probability that a family with 3 children have 3 boys given they have at least 1
boy?
4. (a) Find the truth table for the Boolean Polynomial F(w,x,y,z) = wx’z + xy’
(b) Find the Disjunctive Normal Form of the polynomial in part (a).
(c) Find the Conjunctive Normal Form of the polynomial in part (a).
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CMSC 203 Spring 2003 Examination 3

1. (a) How many license plates can a state produce if the plates can contain 6 characters (from 26 letters and 10 digits) if they can only use one digit? (b) How many ways can Mr. Paul choose 6 students from a class of 15 Boys and 12 Girls, if he must choose at least 5 boys? (c) How many orderings are there of the letters of the word STRAWBERRYALARMCLOCK? (d) How many ways can I seat 12 people around a circular table, if a certain pair of people cannot sit next to one another? (e) How many ways can I fill a box of 50 chocolates from 10 types if I must have at least 1 of each type in the box? 2. Let R be the relation on Z given by R = {( a,b ) | a,b ∈ Z and a ≡ b mod 5}. (a) Prove the R is Reflexive. (b) Prove the R is Symmetric. (c) Prove the R is Transitive. (d) Describe the partition of Z induced by R.

Let S be the relation on {1,2,3,4,5} given as S = {(1,1),(1,3),(1,4),(2,1),(2,4),(3,1),(3,2),(3,3),(3,4),(3,5),(4,2),(4,5),(5,3),(5,4),(5,5)}

(e) Graph S-1. (f) Find MS , the Matrix of S. (g) Find M (^) S o M (^) S -

(h) For the database whose entries form the following table: Make Model Year Engine ID Vehicle ID Color Ford Mustang 1972 A1222 FO13579 Black Ford Fiesta 1989 C54322 FO24245 Yellow Chevy Camaro 1991 754342AH CH172389 Black Chevy Caprice 1989 442355CC CH156738 Yellow Olds Cutlass 1992 ANDU33 OL64332 Blue Olds Cutlass 1992 ANGH28 OL61998 White Volvo P1800 1969 44325XX VO44526 White Volvo 240 1986 53526PD VO64690 Black Volvo 760 1992 578868R VO83529 Blue find the Primary Keys and P (^) 3,

3. (a) For a collection of 80 coins, if 53 are quarters, 15 are quarters from the 1990’s, and 24 are coins from the 1990’s, what is the probability the a coin chosen at random is a quarter or is a coin from the 1990’s?

(b) What is the probability that a family with 3 children have 3 boys given they have at least 1 boy?

4. (a) Find the truth table for the Boolean Polynomial F( w,x,y,z ) = wx’z + xy’ (b) Find the Disjunctive Normal Form of the polynomial in part (a). (c) Find the Conjunctive Normal Form of the polynomial in part (a).

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