Discrete Structures Test 2

Question 1. (5 points) Give a counterexample to disprove the following statement.

The sum of any three consecutive integers is even.

Question 2. (10 points) Write a direct proof for the following statement.

The sum of the cubes of two consecutive integers (e.g., 2³ + 3³ = 8 + 27 = 35) is odd.

Question 3. (15 points) Use the First Principle of Mathematical Induction to prove the following statement for every positive integer n.

Question 4. (10 points) Let

A = { p, q, r, s }

B = { r, t, v }

C = { p, s, t, u }

be subsets of S = { p, q, r, s, t, u, v, w }. Find

a) A C =

b) B - C =

c) A B =

d) C A =

e) (B) =

Question 5. Counting Problems (4 points each -- 12 total)

a) How many three-digit numbers less than 600 can be made using the digits 8, 6, 4, and 2?

b) In one state, automobile license plates must:

• start with two digits (0 to 9), but the first cannot be a zero,
• followed by two upper-case letters (A to Z) ,
• followed by a string of two to four digits (leading zeros are allowed).

c) If a black die and a white die are rolled, how many different rolls are possible? (Assume that the 4-black and 2-white roll is a different roll than the 4-white and 2-black roll)

Question 6. (8 points) Nineteen different mouthwash products make the following claims: 12 claim to freshen breath, 10 claim to prevent gingivitis, 11 claim to reduce plaque, 6 claim to both freshen breath and reduce plaque, 5 claim to both prevent gingivitis and freshen breath, and 5 claim to both prevent gingivitis and reduce plaque.

a) How many products make all three claims?

b) How many products claim to freshen breath but do not claim to reduce plaque?

Question 7. (5 points) How many people must be in a group in order to guarantee that three people in the group have the same birthday? (Don't forget about leap year)

Question 8. (9 points) A publishing company's personnel consists entirely of 6 worker in design, 18 in printing, 5 in sales, 2 in accounting, and 3 in marketing. A committee of 5 people is to be formed.

a) In how many ways can the committee be formed if there is to be one member from each department?

b) In how many ways can the committee be formed if there must be exactly two members from printing?

c) If the committee is selected at random, what is the probability that printing will have at least two representatives?

Question 9. (10 points) A florist has unlimited roses, carnations, lilies, and tulips in stock.

a) How many different bouquets of one dozen flowers can be made?

b) How many different bouquets of one dozen flowers can be made if the customer requests no lilies?

c) How many different bouquets of one dozen flowers can be made if at least two of each kind of flower is included?

Question 10. (10 points) For a family of 4 children, what is the probability of each of the following? (Assume boys and girls are equally likely.)

a) What is the probability of 4 girls?

b) What is the probability of at least one boy?

c) What is the probability of 4 girls given that the first two are girls?