Discrete Mathematics
Test #3
Name:
1. Explain why if seven integers are selected from the first ten positive integers, there must be at least two
pairs of these integers with the sum of eleven.(7pt)
2.
For the following problems the correct answer is given. You must show how this number was
determined.
a) How many strings of five lowercase letters of the English alphabet contain the letters a and
(repetition of letters allowed)? (Solution is 312, 750 ).(7pt)
b
b) How many strings of six lowercase letters of the English alphabet contain the letters a and b
where a is somewhere to the left of b in the string and in this problem no repetition is allowed?
(Solution is 3,825,360 ).(7pt)
3.
A department store contains ten men and fifteen women. Give the correct number for the following
problems.
a) How many ways are there to form a committee with six members if it must have the same number
of men and women? (5pt)
b) How many ways are there to form a committee with six members if it is to have more women than
men (The committee must have at least one man on it.)? (5pt)
4.
Find the probability of the following events.
a) What is the probability that a five card hand does not contain the ace of spades? (6pt)
b) What is the probability that a five card hand contains at least one ace? (6pt)
5.
The number of bacteria in a colony triples every hour.
a) Set up a recurrence relation for the number of bacteria after
n hours have elapsed. (5pt)
b) If 100 bacteria are used to begin a new colony, how many bacteria will be in the colony in
hours? (5pt)
6.
Let
an  4an1  3an2 . Verify that an  3n  1 is a solution to the recursive relation. (7pt)
10
7.
Suppose that | A || B || C | 100,
| A  B | 50, | A  C | 60, | B  C | 40, and
| A  B  C | 175 .How many elements are in A  B  C ?(4pt)
8.
Determine whether the following relations are reflexive, symmetric and/or transitive?
a)
The relation R on
a, b, c where R   a, a  , b, b  ,  c, c  ,  a, b  ,  a, c  , c, b  .(3pt)
b) The relation R on the set of all people where
 a, b  R if and only if a is younger than
b .(3pt)
c)
The relation R on the set of integers where
 a, b  R if and only if a 2  b2 .(3pt)
10. Find the matrix which will represent the relation: R on
only if a | b .(6pt)
1, 2,3, 4,6,12 where  a, b  R if and
11. How many vertices and how many edges do the following graphs have?
a) K6 (2pt)
b)
W5 (2pt)
12. Does there exist a simple graph with five vertices of the following degrees? If so, draw such a graph.
a) 3,3,3,3,3 (3pt)
b)
0,1, 2,3, 4 (3pt)
13. Give an example of the following:
a)
A relation on
b) A relation on
1, 2 that is symmetric and transitive, but not reflexive. (5pt)
1, 2,3 that is reflexive and transitive but not symmetric. (5pt)