Discrete Math Problems

Answer the following questions:

Section 3.1 Homework

1) Suppose a company will select 3 people from a collection of 12 applicants to serve as a regional manager, an assistant regional manager, and an assistant to the regional manager. In how many ways can the selection be made?

2) How many distinguishable permutations can be made of the letters in the word RACECAR?

Section 3.2 Homework

1) From a class of 8 males and 22 females, 2 males and 2 females are needed for the final act of a dance recital. In how many different ways can the 4 dancers be selected?

2) A fair 6-sided dice is rolled 5 times and the result is recorded for each roll. How many different results are possible? Of the possible results, in how many ways can there be a result containing exactly 2 rolls of a 4?

Section 3.3 Homework

1) Show that if 1,343 college freshmen enroll in 48 different sections of an Algebra I course, then 1 course section will have at least 28 students.

Section 3.5 Homework

1) Suppose Jim climbs stairs in a parking garage for exercise. He will sometimes take two steps at a time. Let cn be the number of ways that Jim can climb n steps.
a) Give a recurrence relation for cn. Be sure to include the initial conditions.
b) Use this recurrence relation to calculate in how many ways Jim can climb a flight of 12 steps.

2) Let an = -2an-1 + 15an-2 with initial conditions a1 = 10 and a2 = 70.
a) Write the first 5 terms of the recurrence relation.
b) Solve this recurrence relation.
c) Using the explicit formula you found in part b, evaluate a5. You must show that you are using the equation from part b.

Appendix A, B, C Homework

1) Consider the following sets (30 points):
U = {a, b, c, d, e, f, g}
A = {b, c, d, f}
B = {a, d, g}
C = {a, e}

Represent each of the following with an array of zeros and ones:
a) A ∩ B
b) C ∪ B

2) Consider the following two propositions:
p: It snows tonight.
q: I will stay home.

Use negation (~), conjunction (˄), disjunction (˅), and/or implication (→) to construct a logical equivalence of p→q.
Construct the truth table for both statements and explain how the truth tables establish logical equivalences.

3) Consider the statement “There does not exist a narwhal that can live on land.” Write an equivalent English statement that begins with the words “Every narwhal…(20 points)”
Every narwhal that can not live on land.
Let P(x) be the predicate “x can live on land,” where the variable x represents animals. Write both statements symbolically using P(x) and quantifiers, (∃, ∀). (20 points).

4) Use Bacon’s code to create a dummy message for BURDEN. For the sake of simplicity, use bold font for 0 and regular font for 1. (30 points)

5) At a regional competition, 7 male runners compete in a 100-meter sprint and 5 female runners compete in a separate 100-meter sprint. How many different arrangements are possible for a first-, second-, and third-place male runner and a first- and second-place female runner?

6) How many distinguishable permutations can be made of the letters in the word AMERICA?

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Section 3.1 Homework
1) There are three positions. According to the problem we can select following ways

Regional Manager    Assistant Regional Manager    Assistant to Regional Manager
12 ways                      11 ways                                      10 ways

Thus total number of ways = 12*11*10 = 1320 ways...