1. Write the first five terms of the sequence with 0 3 and. 2. Write an explicit rule and a recursive rule for the sequence.

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1 LESSON 12.1 Name.Date Warm-Up Exercises For use before Lesson 12.1, pages Avnilnbic as a tr«ms(iarency Evaluate. 1. 3! 2. 7! 4! 4. 10! (10-4)! Daily Homework Quiz For use after Lesson 11.5, pages B 1. Write the first five terms of the sequence with 0 3 and an = a»-i Write an explicit rule and a recursive rule for the sequence. a. «= 3, r = 7 b. al ~ 3 d = Write a recursive rule for the sequence 55,11, A lake initially contains 4400 fish. Each year the population declines 30% and the lake is restocked with 300 fish. Write a recursive ule for the number of fish at the beginning of the nth year. How many fish are in the lake at the beginning of the fifth year? Copyright McDougaJ Liftejl Inc All rights reserved lgebra 2

2 E3 Suppose you are playing a game in which you need to choose a vowel and a digit. How many different ways can the vowel and digit be chosen? One way to answer this question is to make a table showing all of the possibilities. There are 50 different ways to choose a vowel and a digit A 0-A 1-A 2-A 3-A 4-A 5-A 6-A 7-A 8-A 9-A R 0-E 1-E 2-E 3-E 4-E 5-E 6-E 7-E 8-E 9-E T o u o-u 1-U 2-U 3-U 4-U 5-U 6-U 7-U 8-U 9-U 1. Suppose you flip a coin and roll a number cube at the same $ time. Use a table to find the number of possible outcomes. 2. Suppose you flip two coins, a penny and a nickel, at the same time. Use a table to find the number of possible outco es. 3. Suppose you roll two number cubes, a blue one and a red one, at the same time. Use a table to find the number of possible outcomes. 4. Suppose you are selling T-shirts that are available in white, green, red, or blue. The sizes are M, L, and XL. How many different T-shirts are possible? 5. Describe any pattern you can observe in the answers to the questions above. Algebra 2 Copyright McDouga! littell Eric All rights reserved

3 LESSON N me Practice A Date Each event can occur in the given number of ways. Find the number of ways all of the eve ts can occur. 1. Event 1: 2 ways, Event 2: 4 ways 2. Event 1: 6 ays, Event 2:1 way 3. Event 1: 7 ways, Event 2: 3 ways 4. Event 1: 2 ways, Event 2: 5 ways, Event 3: 3 ways Fo the gi en configuration, determine how many d ffe ent co puter asswords are possible if (a) digits and letters can be repeated, and (bi digits an letters cannot be repeated. L ss»on 1? digits followed by 2 letters 6. 5 digits followed by 1 letter 7. 4 letters followed by 2 digits 8. 5 letters followed by 1 digit Evaluat the factorial ! 11. 3! Find the number of e mutations Pj P2 Find the numbe of distinguishable permutations of the letters in the word. 17. CAT 18. MONEY 19. UTAH 20. FAMILY 21. MOM 22. TENT 23. PHYSICS 2. FOLLOW 25. Home Decor You are choosing curtains, paint, and carpet for your room. You have 12 choices of curtains, 8 c oices of aint, and 20 choices of carpeting. How many different ways can you choose curtains, paint, and carpeti g for your room? 26. Naming a Dog You are c oosing a name for your registered beagle. Your dog s grandparent s names were Willow-Sutton, Carolina-Downing, Hollybrook-Loner, and Starfire-Wolf. You want your dog s first name to be the same as one of its grandparents first names, an its second name to be the same as one of its grandparents secon names. However, your dog cannot have exactly the same name as one of its grandparents. How many names are possible? 16 Algebra 2 Copyright McDouoal Littell Inc All rights reserved,

4 Name Date Each event can occur in the gi en number of ways. Find the number of ways all of the events can occur. 1. Event 1: 3 ways. Event 2: 4 ways 3. Event 1: 4 ways, Event 2: 6 ways Event 3: 2 ways 2. Event 1: 1 way, Event 2: 5 ways 4. Event 1: 2 ways, Event 2: 9 ways, Event 3: 5 ways Fo the gi en conf guration, determine how many diffe ent computer passwords are possible if (a) digits and letters can be repeate, an (b) digits and lette s cannot be repeated. Loi»son letters followed by 4 digits 6. 1 letter followed by 5 digits 7. 3 digits followed by 3 letters 8. 1 digit followed by 5 letters Evaluate the factorial. 9. 6! ' 11. O' ! Find the number of permutations DP8 14. SPQ 15. 6Pi Find the number of distinguishable permutations of the letters in the word. 17, ENGLISH 18. NORTH 19. MATH 21. EYE 22. ALPHABET 23. OKLAHOMA 25. School Lunch You school cafeteria offers three s l ds, four mai courses, two vegetables, nd three desserts. How many different lunches consisting of a salad, main course, a vegetable, nd dessert re possible? 16. 6P6 20. BELL 24. CALIFORNIA 26. Stacking Boohs Five books are ta en from a shelf and laid in a stack on a table. I how many dif erent orders can the boo s be stacked? 27. Batting Order A baseball coach is determining the batting order for the team. The t am has nine membe s, but the coach does not want the pitcher to be one of the first four to bat. How m ny batting orders are possible? 28. Scheduling Classes Next year you are talcing math, English, h story, keyboarding, chemistry, physics, and physical education. Each class is offered duiing each of the seve p riods in the d y. In how many different orders can you schedule your classes? Copyright McDougal Litteii Inc At! rights reserved Algebra 2 Chapter 12 Resource Boo

5 Name Reteaching with Practice Date Use the fundamental counting principle and ermutations to count the nu be of ways an event can ha pen j Vocabulary Fundamental Counting Principle I If one eve t c n occur in m ways and another event can occur in n I ways, then t e number of ways that both events can occur is m n. his ; principle can be extended to three or more e ents. Lt i»z>on 12 j An ordering of n objects is a permutation of the objects. There are si f permutations of the letters A, B, and C: ABC, ACB, BAC, BCA, CAB, and CBA. 5 Permutations of n Objects aken /* at a Time The number of permutations of r objects taken from a group of n distinct n\ ' [ objects is denoted by npr and is give by: npi = i j Permutations with Repetition ; The number of distinguishable permutations of n objects where one object is repeated q{ times, nother is repeated q2 times, nd so on is n! 4i! * Using the Fundamental Cou ting Principle Radio station call letters consist of four letters beginning with either a K oraw. a. How many different radio station call letters are possible if letters can be repeated1? b. How many different radio station call letters are possible if letters cannot be re eated? Solution a. There are 2 choices for the first letter and 26 c oices for the remaining three letters. Use the fundam ntal counting principle to find the umber of differen possibilities. Number of caluetters = 2 * ,152 b. If you cannot repeat letters, there are still 2 choices for the first letter, but then there are only 25 remaining choices for the second letter, 24 choices for the third letter, and 23 for the fourth letter. Use the funda mental counting principle to find the number of different possibilities. Number of call letters without repetition = 2 * 25 * = 27,600 Copyright McDougal Lrttell Inc ll rights reserved Algebra 2 19

6 ESSON 12.1 CONTINUED Name Date Reteaching with Practice For use with ages Exercises for Example 1 1. A baseball coach is determining the batting order for the team. The team has 9 players, but the co ch does not want the pi cher to be one of the first four to bat. How many batting orders are possible? 2. How many diffe e t 4-digit numbers can be formed from the digits 1, 2, 3, and 4 if digits can be repeated? If digits cannot be repeated? 3. How many different 5-digit zi codes can be formed if digits can be repeated? If digits cannot be repeated? Finding the Numbe of Permutations a. In how many different ways can 2 students out of wenty-fivemember class be elected president and vice president1? b. Find the number of distinguishable permutations of the letters in MATHEMATICS. Solution a. Any of the 25 students can be elected president, then any of the remaining 24 can be elected vice president. So the numbe of ways the students can be elected is 25 * 24 = 600. b. MATHEMATICS has 11 letters of which M is repeated 2 times, A is repeated 2 times, and T is re eated 2 times. Therefo e, the numbe of distinguishable ermutations is given by n\ 11! qi\>q2\-. * qnl 21*21*2! Exercises for Exam le 2 39,916, ,989, If eight basketball teams are in a tou ament, find the number of different ways that first, second, and third lace can be decided. (Assume there are no ties.) 5. There are 15 members in a committee. In how many different ways can a president, vice p esident, secreta y, and treasurer be chosen? 6. Find the number of distinguishable ermutations of the letters in CAT. 7. Find the number of distinguishable permutations of the letters in CINCINNATI. 20 Algebra 2 Copyright McDougaf Littell Inc ll rights re erved.