Discrete math-combinations permutations and probability

Question

Assignment:

Q1. Given a two-step procedure where there are n1 ways to do Task 1, and n2 ways to do Task 2 after completing Task 1, then there are _________ ways to do the procedure.
a. n₁ + n₂
b. n₁ log n₂
c. n₁ * n₂
d. n₁²

Q2. How many bit strings of length 10 begin with 1101?
a. 2¹⁰
b. 2⁶
c. 2⁴
d. None of the above.

Q3. The 10-digit telephone numbering system in North America traditionally defined the Area Code as being NYX, the Office Code as NNX, and the Station Code as XXXX, where N is an integer in the range 2 through 9, Y is either a 0 or 1, and X is an integer from 0 through 9. How many area codes were there?
a. 800
b. 640
c. 480
d. 160

Q4. The telephone numbering system in North America was modified to format individual telephone numbers using NXX-NXX-XXXX, where N is an integer in the range 2 through 9 and X is an integer from 0 through 9. How many different telephone numbers are now possible in North America?
a. 6,400,000,000
b. 4,800,000,000
c. 2,560,000,000
d. 1,024,000,000

Q5. What is the value of k after the following code terminates?
k := 0
for i₁ := 1 to 2
 for i₂ := 1 to 2
  for i₃ := 1 to 2
   for i₄ := 1 to 2
    for i₅ := 1 to 2
     for i₆ := 1 to 2
      for i₇ := 1 to 2
       for i₈ := 1 to 2
        for i₉ := 1 to 2
         k := k + 1
a. 256
b. 512
c. 1,024
d. 2,048

Q6. In a version of programming language BASIC, variable names must be a string of one or two alphanumeric characters, where uppercase and lowercase letters are not distinguished. [Alphanumeric characters are the 26 letters of the English alphabet {A, B,...,Z} plus the digits {0,1,...,9}.] A variable name must begin with a letter. You are offered $1,000 to define a unique variable name to identify each of 1,000 students in a school database. You reasonably decide to:

a. accept the task and begin work because you need $1,000.
b. explain that the goal is mathematically impossible because of your knowledge of discrete math and the product rule.
c. explain that the goal is mathematically possible provided you can use the dollar sign ($) as an alphanumeric character.
d. agree to take on the task, provided you may use the programming language C++, which is not so constrained, to generate the list.

Q7. Limitations on the number of Class A and Class B Internet network numbers (netids) have revealed 32-bit IP addressing to be inadequate to support the future growth of the Internet.
a. True
b. False

Q8. Let A₁ and A₂ be sets. Let T₁ be the task of choosing an element from set A₁, and T₂ be the task of choosing an element from set A2. Therefore, there are |A₁| ways to do T₁.
a. True
b. False

Q9. Let A₁ and A₂ be sets. Let T₁ be the task of choosing an element from set A₁, and T₂ be the task of choosing an element from set A₂. The formula for the union of two sets can be used to determine how many ways you can do both tasks, in any order.
a. True
b. False

Q10. Tree diagrams may be used to solve counting problems.
a. True
b. False

Q11. Using the generalized pigeonhole procedure, if 21 students are placed into 9 classes, then there is at least one class that contains at least __________ student(s).
a. 1
b. 2
c. 3
d. 4

Q12. Applying the pigeonhole principle, with any group of 366 people, at least two must have the same birthday.
a. True
b. False

Q13. Given there are four seasons and 11 public holidays in a year, at least one season must have at least __________ holidays.
a. 0
b. 1
c. 2
d. 3

For questions 14 and 15, use the following information:

The Internet uses a datagram protocol. Each datagram contains a header area organized into a maximum of 14 different fields, as well as a data area that contains the actual data being transmitted. One of the 14 header fields is the header length field (HLEN), which is four bits long and indicates the number of 32-bit blocks that compose the header. Another header field is the 16-bit total length (TLEN) field which specifies the length of the entire datagram, including the header and the data areas. The length of the data area may be computed by subtracting HLEN from TLEN.

Q14. The largest possible value of TLEN determines the maximum total length in octets (blocks of 8 bits) of an Internet datagram. What is this value?
a. 2¹⁶
b. 2¹⁶ - 1
c. 2¹⁶ * 2⁸
d. 2¹⁶ - 2⁸

Q15. The minimum - and most common - value for HLEN is 20 octets. What is the maximum total data area length in octets?
a. 65,536
b. 65,535
c. 65,516
d. 65,515

Q16. Find the number of permutations of the word COEFFICIENT.
a. 11!
b. 11! - 4!
c. 11! / 4!
d. 11! / 2!⁴

Q17. A class consists of 20 seniors and 15 juniors. How many picnic committees of size 5 can be formed?
a. P(20,5)
b. P(35,5)
c. C(20,5)
d. C(35,5)

Q18. A class consists of 20 seniors and 15 juniors. How many picnic committees of size 5 can be formed, given that the committee must contain 3 seniors and 2 juniors?
a. P(20,3) * P(15,2)
b. P(35,5) - P(20,2)
c. C(20,3) * C(15,2)
d. C(35,5) - C(20,2)

Q19. Pascal's identity is the basis for a geometric arrangement of the binomial coefficients in a triangle.
a. True
b. False

Q20. What is the probability that a card drawn from a deck of ordinary playing cards is an ace?
a. 1/52
b. 1/13
c. 4/13
d. 1/4

Q21. How many different 13-card bridge hands can be dealt from a deck of 52 cards?
a. P(52,13)
b. C(52,13)
c. 52! / 13!
d. (52! / 13!) * 39!

Q22. A state lottery commission decides to stimulate interest in gambling by increasing the payoff for their "Pick 6" lottery. To accomplish this feat, they decide to double the minimum jackpot from $1 million to $2 million and to increase the number of numbers from 1 through 40 to 1 through 42. What is your chance of winning if you buy one ticket?
a. 1 / 3,838,380
b. 1 / 5,245,786
c. 1 / 4,496,388
d. 1 / 749,398

Q23. The permutations of the set {1, 2, 3, 4, 5, 6} are arranged in lexicographic order. What is the number following 364125?
a. 364152
b. 364512
c. 364521
d. 364215
Q24. The next bit string procedure produces the next largest bit string after b_n-1b_n-2...b₀ is defined is as follows:
i := 0
while b_i = 1
begin
b_i := 0
i := i + 1
end
b_i := 1
What is the next largest bit string after 10 0011 1111?
a. 10 0011 0000
b. 10 0101 0000
c. 10 0110 0000
d. 10 0100 0000 by calculation

Q25. Find the next largest 4-combination of the set {1, 2, 3, 4, 5, 6} after
{1, 2, 5, 6}.
a. {1, 2, 6, 5}
b. {1, 3, 2, 4}
c. {1, 3, 2, 5}
d. {1, 3, 4, 5}