Counting

How many ways can the squirrel climb this tree to reach a red berry at the top? We could count all the berries on the green leaves: 1, 2, 3, ..., 61, 62, 63. Or we could take a shortcut by counting the green leaves on each tree branch: 7+7+7+7+7+7+7+7+7 = 63. Can you think of a faster way? If so, you are on your way to understanding the Fundamental Principle of Counting. The fundamental counting principle is based on the idea that adding is a shortcut to counting and multiplying is a short cut to adding. This lab, and the next few, are all about learning those shortcuts.

1. Mr. Ollo stared into his closet and found only two pairs of pants—a light pair and a dark pair—that he could still fit into after the semester break. There he also found five shirts that were not in the laundry basket. Mr. Ollo decided to impress his students by wearing a tie. He found six psychedelic ties that truly amazed the eye. How many combinations of pants, shirts, and ties could he wear?
2. You are packing for a weekend trip. You go to your closet and find three shirts and four pants that you can wear. You also find five pairs of shoes in back of your closet. How many combinations of shirts, pants, and shoes can you wear?
3. You are packing for a weekend trip. You go to your closet and find 9 blouses and 6 skirts that you can wear. You also find 12 pairs of shoes in your closet. How many combinations of blouses, skirts, and shoes can you wear?
4. How many ways can you paint a 4-story building if each floor can be only one color, and you only have 2 colors of paint?
5. How many ways can you paint a 5-story building if each floor can be only one color, and you only have 2 colors of paint?
6. How many ways can you paint a 4-story building if each floor can be only one color, and you only have 3 colors of paint?
7. How many ways can a bug walk from the center of a wild flower with 8 stems to the end of a flower petal, if each stem has 13 petals?
8. How many ways can you turn the light switches on and off in your apartment if you have 7 lights?
9. How many ways can you turn the light switches on and off in your apartment if you have 8 lights?
10. How many ways can you turn your light switches on, off, and half-brite if you have 7 lights each with 3 settings?
11. How many ways can you roll three dice? Each die is a cube. Each side of the cube is unique. A "roll" of the dice refers to how the dice land after being thrown.
12. How many ways can you roll three polyhedral dice with 4, 6, and 8 sides, respectively?
13. A dodecahedron has 12 sides. An icosahedron has 20. How many ways can you roll five polyhedral dice with 4, 6, 8, 12, and 20 sides, respectively?
14. How many ways can you answer a True/False test with 10 questions?
15. How many ways can you answer a True/False test with 20 questions?
16. How many ways can you answer a multiple choice test with 10 questions if each question has 4 choices?
17. How many ways can you answer a multiple choice test with 10 questions if each question has 5 choices?
18. Regular Tennessee license plates contain 3 letters and 3 digits. What is the maximum number of vehicles that can be licensed in Tennessee?
19. In 1959, regular Tennessee license plates contained 2 letters and 3 digits. What was the maximum number of vehicles that could be licensed in Tennessee in 1959?
20. Regular California license plates contain 3 letters and 4 digits. What is the maximum number of vehicles that can be licensed in California?
21. Tennessee personalized auto license plates may not have less than three nor more than seven numbers, letters or combinations thereof. What is the maximum number of Tennessee personalized auto license plates with 3 numbers and/or letters?
22. Tennessee personalized auto license plates may not have less than three nor more than seven numbers, letters or combinations thereof. What is the maximum number of Tennessee personalized auto license plates with 4 numbers and/or letters?
23. Tennessee personalized auto license plates may not have less than three nor more than seven numbers, letters or combinations thereof. What is the maximum number of Tennessee personalized auto license plates with 5 numbers and/or letters?
24. Tennessee personalized auto license plates may not have less than three nor more than seven numbers, letters or combinations thereof. What is the maximum number of Tennessee personalized auto license plates with 6 numbers and/or letters?
25. Tennessee personalized auto license plates may not have less than three nor more than seven numbers, letters or combinations thereof. What is the maximum number of Tennessee personalized auto license plates with 7 numbers and/or letters?
26. How many pizzas can you make if you have 4 toppings (pepperoni, peppers, olives, and sausage) from which to choose? Assume that all have sauce and cheese on them. Assume that half pepper and half pepperoni is the same as peppers and pepperoni.
27. Suppose you run out of olives, how many pizzas can you make? (Read the problem above.)
28. Suppose you add mushrooms to your list of toppings. How many pizzas can you make? (Read the problem above.)
29. Suppose you had 20 different toppings, how many different pizzas can you make? (Read the problem above.)
30. The DNA molecule is made up of four kinds of smaller molecules. These smaller molecules are often referred to by the letters A, C, G, T, which symbolize the bases adenine, cytosine, guanine, and thymine. Groups of these small molecules along the DNA molecule make up the code. How many groups of 3 DNA molecules are there?
31. How many groups of 4 DNA molecules are there? (Read the problem above.)
32. How many groups of 4 DNA molecules are there that start wth the letter C? (Read the problem above.)
33. How many ways can you make DNA with 4 bases G A C T and 6 billion slots to place them in?
34. How many ways can you take DNA strands of 6 acids in length from an alien race whose DNA consists of 5 amino acids?
35. How many ways can you take DNA strands of 5 acids in length from an alien race whose DNA consists of 6 amino acids?
36. The keys of General Motors cars have six parts. Originally, General Motors used two patterns for each part. How many different key designs were possible?
37. The keys of General Motors cars have six parts. If General Motors uses three patterns for each part, how many different key designs were possible?
38. The keys of General Motors cars have six parts. If General Motors uses four patterns for each part, how many different key designs were possible?
39. How many ways can you choose 3 of the 24 Greek letters for a fraternity or sorority name?
40. How many ways can you choose 3 of the 24 Greek letters for a fraternity or sorority name if each letter must be different?
41. How many ways can you elect a committee of 50 people if each state may appoint either its governor or one of its two senators?
42. How many ways can you assign a user ID with 8 characters if it can contain only letters, digits, and underscores?
43. How many ways can you assign a user ID with 8 characters if it can contain only letters, digits, and underscores, and the first character may not be a digit?
44. How many ways can you make 7 digit phone numbers if the first digit cannot be a 0 or a 1?
45. In the original plan for area codes for phones, the first digit could be any number from 2 through 9, the second digit was either 0 or 1, and the third could be any number except 0. According to this plan, how many different area codes were possible altogether?
46. How many ways can you assign cell phone numbers if each number has 10 digits in this form: NXX-NXX-XXXX? The first 3 digits are the country's prefix. The X represents any digit from 0 thru 9. The N represents any digit from 2 thru 9. Could everyone alive now have their own phone number?
47. The Looloolo language consists of only two letters: 'L' and 'O', but every arrangement is a legitimate word. How many ways 8 letter Looloolo words are there? Looloolo is one of them.
48. The Loopoolopo language consists of only three letters: 'L', 'O' and 'P', but every arrangement is a legitimate word. How many ways 8 letter Loopoolopo words are there? Loopoolopo is one of them.
49. How many ways can a team win or lose the first four games of a World Series?
50. How many ways can you choose an outfield if you have 5 left fielders, 7 right fielders, and 3 center fielders?
51. How many ways can you choose a backfield if you have 6 halfbacks, 4 fullbacks, and 3 quarterbacks
52. How many ways can you put into play a basketball team if you have 6 guards, 4 forwards, and 3 centers?
53. The Bach Stradivarius trumpet includes six different flares, each of which is available in light, regular and heavy weight and your choice of yellow or gold brass. In addition, there are 8 choices for the mouthpiece and 5 choices for the bore. How many different trumpets are possible from these choices?
54. In how many ways can a casting director choose a mother, a father, and one child, from two actresses, three actors and four children?
55. How many ways can you order a drink from an ice cream parlor that offers three drinks, sodas, milk shakes, and blasters in three sizes, small, medium, and large, with 24 flavors. If you order one drink, how many choices do you have?
56. How many ways can you order a drink from an ice cream parlor that offers three drinks, sodas, milk shakes, and blasters in three sizes, small, medium, and large, with 31 flavors. If you order one drink, how many choices do you have?
57. How many different flavors of ice cream must you buy in order to make at least ten different two-scoop cones if vanilla-strawberry is different from strawberry-vanilla?
58. Now suppose that vanilla-strawberry is the same as strawberry-vanilla. How many different flavors of ice cream must you buy in order to make ten different two-scoop cones?
59. How many different flavors of ice cream must you buy in order to make at least ten different two-scoop cones if double scoops of any flavor are counted?
60. Banana splits require 3 different flavors of ice cream. Assume that vanilla-chocolate-strawberry is the same as chocolate-strawberry-vanilla, but different from vanilla-chocolate-pistachio. How many different flavors of ice cream must you buy in order to make ten different banana splits?

1. Answer question 1 using the fundamental principle of counting.
2. Check your answer to question 1 by listing all the possibilities.
   Hint: use letters, digits, or symbols for each item in Mr. Ollo's wardrobe.
3. Draw a tree diagram for question 1.
4. Answer the pizza problems in questions 26 thru 29.
5. Answer question 26 using the fundamental principle of counting.
6. Check your answer to question 26 by listing all the possibilities.
7. Draw a tree diagram for question 26.