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Name: Date: Algebra 2 Counting Review Objective: You will correctly solve varied review problems involving counting. Review problems 1. a. Based on the symmetry relationship in Pascal’s Triangle, what other combination number has the same value as 9C5? b. Based on the adding relationship in Pascal’s Triangle, what are the two combination numbers that can be added to get 9C5? c. Show how to find the value of 9C5 using arithmetic (not using a calculator shortcut, and not using Pascal’s Triangle). Show your work. d. Find the value of 9C5 again, by writing Pascal’s Triangle from the beginning through the row that contains 9C5. e. Write a counting problem of your own, whose answer is given by 9C5. Name: Date: Algebra 2 2. a. Identify another combination number that is equal to b. What single combination number is equal to 17C5 12C7 . + 17C6 ? c. What is the easiest way to find the sum 10C0 + 10C1 + 10C2 + 10C3 + 10C4 + 10C5 + 10C6 + 10C7 + 10C8 + 10C9 + 10C10 ? Directions for problems: First write the calculation that needs to be done, then write the answer. Using calculator shortcuts is allowed, but you still must show what calculation you are doing. If you only write the final answer, you will not get full credit. For example, first write 5 · 4, then write 20. Or, first write 5P2, then write 20. Don’t just write 20. 3. Out of these 6 school subjects English, Social Studies, Math, Science, Foreign Language, Physical Education you are asked to choose your 1st favorite, 2nd favorite, and 3rd favorite subjects. How many different ways could you make these choices? 4. There are 36 students on the high school Senate. Suppose that two of these students must be elected to the positions of Moderator and Secretary. How many different ways could the election turn out? 5. There are 60 homerooms in the school. Suppose that 5 lucky homerooms will be randomly chosen to receive pizza parties. How many different ways could the pizza parties be awarded? 6. The letters APRIL can be rearranged in various ways: ARILP, RPLIA, LIPAR, etc. How many different ways can this be done? Name: Date: Algebra 2 7. A push-button lock has five buttons that look like this: [1] [2] [3] [4] [5] A lock combination is a sequence of six button presses (examples: 231154; 315232; 444444). How many different lock combinations are possible? 8. Suppose you are ordering a salad. You have the choice of whether or not to include each of these items in your salad: tomato, cucumber, carrot, onion, radish, croutons, dressing How many different ways could you order your salad? 9. There are 5 multiple-choice questions with choices (A), (B), and (C). How many different ways could this set of questions be answered? 10. An ATM (banking machine) requires a 4-digit PIN (password number). The PIN can be any four digits (examples: 0123, 5555). How many different PIN’s are there? 11. Suppose that next week, it rains on 3 out of the 7 days. How many different possibilities exist for the set of days that it rains? Name: Date: Algebra 2 12. From 12 available student-athletes, a track coach must pick 4 different people to compete in the long jump, the high jump, the javelin throw, and the hurdles. How many different ways could the coach make this set of choices? 13. You flip a coin 7 times. How many possible coin flip sequences have exactly 2 heads and 5 tails (example: HTTTTHT)? 14. Suppose you are ordering a salad. You have the choice of whether or not to include each of these 7 toppings in your salad: tomato, cucumber, carrot, onion, radish, croutons, dressing a. How many different ways could you order your salad? b. How many possible salads have exactly 4 toppings? 15. TGI Friday’s Restaurant has a special deal to get an appetizer, entrée and dessert all for one price. Below are the options for each course of the meal. Appetizer Entrée Dessert Classic Wedge Salad Parmesan-Crusted Chicken Chocolate Peanut Butter Pie Fried Mozzarella Chicken Fingers Vanilla Bean Cheesecake Tuscan Spinach Dip Cheeseburger Pot Stickers Brushetta Chicken Pasta Shrimp Key West Petite Sirloin How many different ways could you order this deal? Name: Date: Algebra 2 Answers to today’s review problems 1. a. 9C4 b. 8C4 + 8C5 c. 9C5 = 9 87 6 5 = 126 1 2 3 4 5 d. Triangle should include this row. The underlined value is 9C5. 1, 9, 36, 84, 126, 126, 84, 36, 9, 1 e. Any problem involving choosing 5 out of 9 things without an order. 2. a. 12C5 b. 3. 6P3 = 120 4. 36P2 = 1260 5. 60C5 = 5,461,512 6. 5! = 120 7. 56 = 15625 8. 27 = 128 9. 35 10. 104 11. 7C3 12. 12P4 13. 7C2 14. a. 27 b. 7C4 15. 4 * 6 * 2 18C6 c. 210