(a) A Chinese restaurant offers 10 different lunch specials. Each weekday for one week, Fiona goes to the restaurant and selects a lunch special. How many different ways are there for her to select her lunches for the week? Note that which lunch she orders on which day matters, so the following two selections are considered different. One possible selection: Mon: Kung pao chicken Tues: Beef with broccoli Wed: Kung pao chicken Thurs: Moo shu pork Fri: Beef with broccoli A different selection: Mon: Beef with broccoli Tues: Kung pao chicken Wed: Kung pao chicken Thurs: Moo shu pork Fri: Beef with broccoli Counting strings over {a, b, c} (c) The string contains at least 8 consecutive identical characters. (d) The first character is the same as the last character, or the last character is a, or the first character is a. Counting calculus students. (b) You are given the following data about Group B. 28 students in Group B have taken Math 2A, 28 have taken Math 2B, and 25 have taken Math 2C. 11 have taken both Math 2A and 2B, 9 have taken both Math 2A and 2C, and 10 have taken both Math 2B and 2C. 3 have taken all three classes. How many students are in Group B? Counting ways to line up for a family photo. A family lines up for a photograph. In each of the following situations, how many ways are there for the family to line up so that the mother is next to at least one of her daughters? (a) The family consists of two parents, two daughters and two sons. (b) The family consists of two parents, three daughters and four sons. Counting passwords without repeating characters. Consider the following definitions for sets of characters: Digits = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } Letters = { a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z } Special characters = { *, &, $, # } Compute the number of passwords that satisfy the given constraints. (b) Strings of length 6. Characters can be special characters, digits, or letters, with no repeated characters. The first character can not be a special character. Strings with no repetitions. How many strings are there over the set {a, b, c} that have length 10 in which no two consecutive characters are the same? For example, the string "abcbcbabcb" would count and the strings "abbbcbabcb" and "aacbcbabcb" would not count. The following picture was taken at the Acton Science Discovery Museum. Answer the following questions based on this: I go to a big ice cream shop. They have 20 ice cream flavors (including chocolate), 5 sorbet flavors, and 6 froyo flavors. Answer the following for the big ice cream shop instead of just the 4 flavors at the children's museum. Instructions: Show your work and label each step as Rule of Sums or Products. 1. How many two scoop cones are possible? Hint: Decide bottom scoop, then decide top scoop. Note: Chocolate on bottom, vanilla on top is a different arrangement than vanilla on bottom, chocolate on top. 2. How many two scoop cones contain chocolate? Answer using Inclusion-Exclusion 3. How many two scoop cones contain two different flavors? 4. How many three scoop cones are possible? 5. How many three scoop cones contain chocolate? Answer using Inclusion-Exclusion 6. How many ways can I get a cone with two or three scoops? \f1. Use the photo above to answer a these questions about the ice cream shop with 31 flavors: o a) How many 3 scoop cones contain 3 different flavors? Answer twice: Once if we only care which flavors there are. Second if we care the order in which flavors are scooped. o b) How many 3 scoop cones contain exactly 2 flavors? Answer twice: Once if we only care how much of each flavor we have. Second if we care the order in which flavors are scooped. 2. Now some questions about cards (normal 52 card deck with 4 suits, 13 cards each): o a) How many 5 card hands contain 3 Kings and 2 fours? A "hand" is usually defined as 5 cards, with no particular order. o b) How many 5 card hands contain at least 1 King? Count by complement. o c) How many 5 card hands contain at least 2 Kings? Answer twice: First, count by complement. Second, add up the cases of 2, 3, 4, 5 Kings. \f1. Use the photo above to answer a these questions about the ice cream shop with 31 flavors: o a) How many 3 scoop cones contain 3 different flavors? Answer twice: Once if we only care which flavors there are. Second if we care the order in which flavors are scooped. o b) How many 3 scoop cones contain exactly 2 flavors? Answer twice: Once if we only care how much of each flavor we have. Second if we care the order in which flavors are scooped. 2. Now some questions about cards (normal 52 card deck with 4 suits, 13 cards each): o a) How many 5 card hands contain 3 Kings and 2 fours? A "hand" is usually defined as 5 cards, with no particular order. o b) How many 5 card hands contain at least 1 King? Count by complement. o c) How many 5 card hands contain at least 2 Kings? Answer twice: First, count by complement. Second, add up the cases of 2, 3, 4, 5 Kings. \f1. Use the photo above to answer a these questions about the ice cream shop with 31 flavors: o a) How many 3 scoop cones contain 3 different flavors? Answer twice: Once if we only care which flavors there are. Second if we care the order in which flavors are scooped. o b) How many 3 scoop cones contain exactly 2 flavors? Answer twice: Once if we only care how much of each flavor we have. Second if we care the order in which flavors are scooped. 2. Now some questions about cards (normal 52 card deck with 4 suits, 13 cards each): o a) How many 5 card hands contain 3 Kings and 2 fours? A "hand" is usually defined as 5 cards, with no particular order. o b) How many 5 card hands contain at least 1 King? Count by complement. o c) How many 5 card hands contain at least 2 Kings? Answer twice: First, count by complement. Second, add up the cases of 2, 3, 4, 5 Kings. \f1. Use the photo above to answer a these questions about the ice cream shop with 31 flavors: o a) How many 3 scoop cones contain 3 different flavors? Answer twice: Once if we only care which flavors there are. Second if we care the order in which flavors are scooped. o b) How many 3 scoop cones contain exactly 2 flavors? Answer twice: Once if we only care how much of each flavor we have. Second if we care the order in which flavors are scooped. 2. Now some questions about cards (normal 52 card deck with 4 suits, 13 cards each): o a) How many 5 card hands contain 3 Kings and 2 fours? A "hand" is usually defined as 5 cards, with no particular order. o b) How many 5 card hands contain at least 1 King? Count by complement. o c) How many 5 card hands contain at least 2 Kings? Answer twice: First, count by complement. Second, add up the cases of 2, 3, 4, 5 Kings