i need help with the numbers 2, 8, 12, today on a doc document please by 9pm just the nswers.

1. A bank password consists of two letters of the English al- tails are possible? phabet followed by two digits. How many different pass- 4. A catered menu is to include a soup, a main course, a words are there? dessert, and a beverage. Suppose a customer can select 2. In a psychological experiment, a person must arrange a from four soups, five main courses, three desserts, and two square, a cube, a circle, a triangle, and a pentagon in a beverages. How many different menus can be selected? row. How many different arrangements are possible? 5. A fair six-sided die is tossed four times and the numbers 3. A coin is tossed four times and the result of each toss is shown are recorded in a sequence. How many different recorded. How many different sequences of heads and sequences are there? 96 Chapter 3 Counting 6. Let A = {0, 1). A* denotes the set of all finite sequences 26. Give a proof of your result for Exercise 25. of elements of A as defined in Section 1.3. 27. A bookshelf is to be used to display six new books. Sup- (a) How many strings of length three are there in A*? pose there are eight computer science books and five (b) How many strings of length seven are there in A*? French books from which to choose. If we decide to show 7. (a) Compute the number of strings of length four in the four computer science books and two French books and set corresponding to the regular expression (01)*1. we are required to keep the books in each subject together, how many different displays are possible? (b) Compute the number of strings of length five in the set corresponding to the regular expression (01)*1. 28. Three fair six-sided dice are tossed and the numbers showing on the top faces are recorded as a triple. How 8. Compute each of the following many different records are possible? (a) 4PA (b) 6 Ps (C) 7P2 29. Prove that n . "-1 Pm-1 = a Pa. 9. Compute each of the following. 30. Most versions of Pascal allow variable names to consist (a) a Pa-1 (b) " Pa-2 (c) n+1 Pa-1 of eight letters or digits with the requirement that the first character must be a letter. How many eight-character vari- In Exercises 10 through 13, compute the number of permuta- able names are possible? tions of the given set. 31. Until recently, U.S. telephone area codes were three-digit 10. {r, s, t, u } 11. {1, 2, 3, 4, 5 } numbers whose middle digit was 0 or 1. Codes whose 12. {a, b, 1, 2, 3, c) 13. {4, 7, 10, 13} last two digits are I's are used for other purposes (for ex- ample, 911). With these conditions how many area codes In Exercises 14 through 16, find the number of permutations of were available? A taken r at a time. 32. How many Social Security numbers can be assigned at 14. A = {1, 2, 3, 4, 5, 6, 7), r = 3 any one time? Identify any assumptions you have made. 15. A = [a, b, c, d, e, f), r = 2 33. How many zeros are there at the end of 12!? at the end of 16. A = {x | x is an integer and x-