Counting Techniques Class Exercises

[Pages:4]MAT 17: Introduction to Mathematics Counting Techniques Class Exercises

Product Rule

1. A restaurant offers a dinner special consisting of an appetizer, a salad, an entr?e, and a dessert. If the restaurant offers 6 appetizers, 4 salads, 10 entr?es, and 5 desserts, how many unique dinner specials can be created?

2. A secret code consisting of five parts is randomly generated by a computer. The first part is a capital letter in the English alphabet. The second part is a color of the rainbow. The third part is a single digit. The fourth part is the name of a month. The fifth part is one of the suits in a standard deck of playing cards. How many unique codes are possible?

Sum Rule

3. How many ways can a person randomly select the name of a month or a day of the week?

4. Determine the number of ways an individual can roll a die or select a card from a standard deck of playing cards.

Subtraction Rule (Inclusion-Exclusion Principle)

5. Determine the number of ways an individual can select a heart or a face card from a standard deck of playing cards.

6. How many ways can a pair of dice be rolled so the total shown is even or a multiple of 5?

7. How many people are in a group consisting of 75 people who work full time, 80 people who attend college, and 30 people who work full time while attending college?

8. A group of 500 people consists of 250 who like math, 200 who like science, and 150 who do not like math or science. How many people in the group like both math and science?

Tree Diagram

9. Determine the number of ways to form a three-digit number consisting solely of prime numerals if the first digit must be less than 4 and no two consecutive digits can be the same or divisors of 15.

Fill-in-the-Blanks Technique

10. How many four-digit counting numbers with no repeated digits exist?

11. How many five-digit counting numbers with a 7 in the hundreds place and no repeated digits exist?

12. How many odd five-digit counting numbers with no repeated digits exist?

Prof. Fowler

13. How many even five-digit counting numbers with no repeated digits exist?

14. How many four-digit counting numbers greater than 3999 with no repeated digits exist?

15. How many even four-digit counting numbers greater than or equal to 7500 with no repeated digits exist?

16. There are five red, six white, four blue, and three green balls in a box. How many ways can one ball be placed on each of four pedestals if the third pedestal must have a red ball, and the fourth pedestal cannot have a red ball?

17. How many ways can the four aces in a standard deck of playing cards be distributed among three students?

18. How many ways can the four aces in a standard deck of playing cards be distributed among three students if the first student must have the ace of spades, but the second student cannot have the ace of clubs?

19. How many ways can a group of eight men and three women be seated if they must be seated in a row such that they alternate two men followed by one woman?

Exponentiation

20. How many ways can one letter grade (A through F only; no plus or minus) be assigned to each of ten students in a math class?

21. Each letter of the English alphabet is on its own slip of paper in a hat. How many five-letter "words" can be formed by randomly selecting slips of paper from the hat if each letter drawn is written down and the slip returned to the hat before the next selection is made?

22. How many ways can a student randomly guess the answers to a ten-question true/false quiz?

23. How many ways can a student randomly guess the answers to a ten-question multiple choice quiz if each question has possible answers of (a), (b), (c), and (d)?

Factorials

24. How many ways can five unique textbooks be arranged on a shelf?

25. How many ways can eight college students line up for dessert at a sundae bar?

26. Determine the number of unique arrangements of the letters of the English alphabet.

27. How many ways can four tigers, six elephants, and five dogs line up to march into the big top at a circus if all members of each species must be grouped together?

28. Determine the number of ways five freshman, three sophomores, six juniors, and seven seniors can be seated in a straight line if all members of each class must sit together.

Prof. Fowler

Permutations

29. Determine the number of ways three from a set of eight unique mathematics textbooks can be arranged on a shelf.

30. Each letter of the English alphabet is on its own slip of paper in a hat. How many five-letter "words" can be formed by randomly selecting slips of paper from the hat if the slips of paper are placed on a table in the order drawn?

31. How many unique five-letter "words" can be formed using only the letters in the phrase "MATH RULES" if each letter can be used only once?

Combinations

32. How many ways can a committee of three people be selected from a group of eight people?

33. Determine the number of unique five-card poker hands that exist in a standard deck of cards.

34. A drawer contains 14 black, 10 blue, and 6 green socks. How many different "pairs" of socks can be pulled out randomly if a "pair" does not have to have matching colors?

35. A drawer contains 14 black, 10 blue, and 6 green socks. How many different pairs of socks can be pulled out randomly if a pair must match in color?

36. How many ways can a scientist select 23 of the 25 rats in a cage to conduct an experiment?

37. How many ways can a scientist randomly select at least 23 of the 25 rats in a cage to conduct and experiment?

38. Snow White and the Seven Dwarves organize a campaign against the use of calculators in school. They want to form a committee of three of them to lobby for their cause. How many such committees can be formed if Snow White must be on the committee?

39. Determine the number of ways a group of four students can be chosen from a group of seven male and three female students.

40. Determine the number of ways two boys and two girls can be randomly selected from a group consisting of seven male and three female students.

41. How many ways can three white and two blue hats be selected from a pile consisting of eight red, eleven white, and twelve blue hats if each hat remains out of the pile once selected?

42. A bag of fifteen apples contains six rotten apples. If a sample of five apples is randomly selected from the box, how many such samples contain two rotten apples?

43. A box of twelve iPhones is dropped during shipping, and three iPhones were damaged as a result. A sample of four iPhones is inspected to determine how bad the damage is. How many such samples contain no more than two damaged iPhones?

Prof. Fowler

44. How many poker hands are a flush (i.e., all the same suit)? 45. How many poker hands contain a pair of threes, a pair of nines, and a face card? 46. How many poker hands are classified as two pair? 47. How many poker hands contain exactly three spades? 48. How many poker hands are a full house (i.e., three cards of one denomination and two cards

of a different denomination)? 49. How many ways can exactly two heads can be obtained on twelve flips of a fair coin? 50. How many ways can at most two heads be obtained on twelve flips of a fair coin? 51. How many ways can at least two heads be obtained on twelve flips of a fair coin? 52. The letters of the English alphabet are written on individual slips of paper and placed in a hat.

Determine the number of ways exactly two vowels can be obtained if six letters are randomly selected without replacement. Permutations with Indistinguishable Objects (Arrangements with Multiple Occurrences of Objects) 53. Determine the number of unique arrangements of the letters in the word "MISSISSIPPI." 54. Determine the number of unique arrangements of the letters in "ABRACADABRA." 55. Determine the number of ways to arrange the letters in the phrase "BANANA CABANA." Pigeonhole Principle (Dirichlet Drawer Principle; Worst-Case Scenario) 56. How many people must be in a group to guarantee at least two of them share a birth month? 57. How many times must a pair of dice be thrown to guarantee some total shown on the dice occurs at least twice? 58. How many times must a 1-to-8 spinner be spun to guarantee some number is spun ten times?

Prof. Fowler

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download