# Combinations and Permutations Problem Set

Question 1:

From a total of 15 people, three committees consisting of 3, 4, and 5 people, respectively, are to be chosen.

b) How many such sets of committees are possible if there is no restriction on the number of committees on which a person may serve?

Question 2:

c) In how many ways can we select a set of five microprocessors containing exactly two defective microprocessors?

d) In how many ways can we select a set of five microprocessors containing at least one defective microprocessor?

Question 3:

In how many ways can 15 identical computer science books and 10 identical mathematics books be distributed among five students?

Question 4:

A major software company is arranging a job fair with the intention of hiring 10 recent graduates. No candidate can receive more than one offer. In response to the company's invitation, 150 candidates have appeared at the fair.

c) How many ways are there to extend job offers to 10 of the 150 candidates, if we already know that at least one of Ron Smart and Fran Wise gets an offer?

d) How many ways are there to distribute the 150 resumes to three interviewers?

e) How many ways are there for three interviewers to select three resumes (one resume for each interviewer) from the pile of 150 resumes for the first interview round?

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#### Solution Preview

Question 1:

b) Since a person may serve multiple committees,

For the first committee of 3 people, there are C(15,3) ways;

For the second committee of 4 people, there are still C(15,4) ways rather than C(12,4) since a person can serve multiple committees;

For the third committee of 5 people, there are still C(15,5) ways.

Since these three committees are independent, the total ways: C(15,3)C(15,4)C(15,5).

Question 2:

c) 95!/(92!*3!) * 5!/(3!*2!)

It is 95!/(92!*3!)*5!/(3!*2!)=95*94*93/(3*2*1)*(5*4)/(2*1)= 1384150

d) This is a complement event. Put in another way, we first figure out the total ways of selecting 5 microprocessors and in ...

#### Solution Summary

The solution discusses the combinations and permutations problem set.

Discrete Math : Combinations, Permutations and Probability (25 MC Problems)

This test consists of 25 equally weighted questions.

1. 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

2. How many bit strings of length 10 begin with 1101?

a. 210

b. 26

c. 24

d. None of the above.

3. 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

4. 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

5. 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

6. 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.

7. 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

8. 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

9. 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

10. Tree diagrams may be used to solve counting problems.

a. True

b. False

11. 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

12. Applying the pigeonhole principle, with any group of 366 people, at

least two must have the same birthday.

a. True

b. False

13. 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.

14. 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

15. 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

16. Find the number of permutations of the word COEFFICIENT.

a. 11!

b. 11! - 4!

c. 11! / 4!

d. 11! / 2!4

17. 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)

18. 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)

19. Pascal's identity is the basis for a geometric arrangement of the

binomial coefficients in a triangle.

a. True

b. False

20. 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

21. 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!

22. 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

23. 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

24. The next bit string procedure produces the next largest bit string after

bn-1bn-2...b0 is defined is as follows:

i := 0

while bi = 1

begin

bi := 0

i := i + 1

end

bi := 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

25. 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}