Math 3201 Assignment 2 Unit 2 Counting Methods Name: Fundamental Counting Principle 2.1 Page 66 [And = *, Or = +] Identify the choice that best completes the statement or answers the question. Show all necessary workings for multiple choice. NO MARKS WILL BE GIVEN FOR ANSWERS WITHOUT APPROPRIATE WORKINGS. 1. Eve can choose from the following notebooks: lined pages come in red, green, blue, and purple graph paper comes in orange and black How many different colour variations can Eve choose if she needs one lined notebook and one with graph paper? A. 6 B. 8 C. 12 D. 16 2. A combination lock opens with the correct three-digit code. Each wheel rotates through the digits 1 to 8. How many different three-digit codes are possible? A. 24 B. 64 C. 512 D. 1024 3. A combination lock opens with the correct three-digit code. Each wheel rotates through the digits 1 to 8. Suppose each digit can be used only once in a code. How many different codes are possible when repetition is not allowed? A. 21 B. 63 C. 256 D. 336 4. How many possible ways can you draw a single card from a standard deck and get either a heart or a club? A. 2 B. 13 C. 14 D. 26 5. Evaluate. 8! + 1! A. 40 321 B. 5041 C. 40 123 D. 16 777 217 6. Evaluate. Factorials and Permutations 2-2 Page 81 A. 0 B. 1 C. 3 D.
7. Identify the expression that is equivalent to the following: HW Page 82 A. N B. n C. n 2 D. n 3 8. Identify the expression that is equivalent to the following: A. B. C. n 2 D. n! 9. Solve for n, where n N. HW Page 82 A. 13 B. 15 C. 17 D. 18 10. Solve for n, where n I. A. 8 B. 9 C. 10 D. 11 Permutations 2-3 [without or with conditions see notes HW Page 93] 11. How many different permutations can be created when 7 people line up to buy movie tickets? A. 49 B. 128 C. 720 D. 5040 12. Evaluate. 21P 2 A. 441 B. 420 C. 399 D. 2 097 152 13. Suppose a word is any string of letters. How many three-letter words can you make from the letters in REGINA if you do not repeat any letters in the word? A. 20 B. 16 C. 216 D. 120
14. How many ways can 7 friends stand in a row for a photograph if Sheng always stands beside his girlfriend? [Items together treat as one unit] A. 1440 B. 5040 C. 360 D. 720 15. How many ways can 8 friends stand in a row for a photograph if Molly, Krysta, and Simone always stand together? [Items together treat as one unit] A. 1440 B. 4320 C. 5040 D. 2160 Permutations when Objects are Identical (Repetition) 2.4 HW Page 104 n! abc!!! 16. How many different arrangements can be made using all the letters in CANADA? A. 120 B. 180 C. 360 D. 720 17. How many different permutations can be made from the word STATISTICS? A. 3628800 B. 604800 C. 100800 D. 50400 [Combinations 2-6 HW Page 118] 18. There are 14 members of a student council. How many ways can 4 of the members be chosen to serve on the dance committee? (No Order!) A. 1001 B. 2002 C. 6006 D. 24 024 19. There are 14 members of a student council. How many ways can 7 of the members be chosen to serve on the dance committee? A. 1144 B. 1716 C. 3432 D. 17 297 280 20. Evaluate. A. 130 B. 126 C. 122 D. 118
21. Evaluate. A. 0 B. 1 C. 11 D. 22 Combinations with Conditions and FCP..HW Page 119 22. Suppose that 3 teachers and 6 students volunteered to be on a graduation committee. The committee must consist of 1 teacher and 2 students. How many different graduation committees does the principal have to choose from? A. 45 B. 60 C. 90 D. 180 22. Suppose that 10 teachers and 8 students volunteered to be on an environmental action committee. The committee must consist of 2 teachers and 2 students. How many different environmental action committees does the principal have to choose from? A. 45 B. 73 C. 1260 D. 5040 23. Identify the term that best describes the following situation: Determine the number of arrangements of six friends waiting in line for movie tickets. A. Permutations B. Combinations C. Factorial D. none of the above 24. Identify the term that best describes the following situation: Determine the number of pizzas with 4 different toppings from a list of 40 toppings. A. Permutations B. Combinations C. Factorial D. none of the above 25. Ten friends on bicycles find an empty bike rack with exactly ten spots for bikes. How many ways can the ten bikes be parked so that Elsa and Rashid are parked next to each other? A. 362 880 B. 725 760 C. 2 177 280 D. 2 000 000 000 26. Ten friends on bicycles find an empty bike rack with exactly ten spots for bikes. How many ways can the ten bikes be parked so that Elsa is at one end of the bike rack? A. 9! B. C. 10! D.
27. Evaluate. A. 330 B. 660 C. 990 D. 1320 28. How many different routes are there from A to B, if you only travel south or east? A. 16 B. 24 C. 28 D. 56 Fundamental Counting Principle Part II Show All Workings 1 In Newfoundland and Labrador, a license plate consist of a letter-letter-letter-digit-digit-digit A) How many license plates are possible if repetition is permitted (no restrictions)? B) How many license plates are possible if repetition is not permitted? C) How many license plates are possible if the first letter must be a C or A and the last digit is 9, if repetition is allowed? D) How many license plates are possible if the first letter must be a C or A and the last digit is 9, if repetition is not allowed?
2 Draw a tree diagram to display and list all possible outcomes for the rolling of a single di, the tossing of a dime and the flipping of a quarter. Use it to determine the number of ways you can get a number greater than three on the di and get heads on the dime and the quarter. 3 How many ways can a teacher seat 6 girls and 4 boys in a row 10 seats, if a boy must sit at the end of each row? Factorial Notation and Permutations 4 Evaluate without a calculator (Show ALL workings!) 102! 300! 8! A) B) C) D) 10 P2 E) 6 P 6 100!101 299!2 5!3! \ n! 2( n 3)! 5 Solve for n: n N A) 182 B) 180 C) n P2 30 ( n 2)! ( n 1)!
Permutations with No Conditions n P r 6 A) How many ways can you select a committee that must have a president, vice president and treasurer from a group of 12 people? [Use npr as well as the FCP] B) A coach has 15 players to choose from. i) How many ways can he select 5 people to play on a starting lineup for a basketball team (left guard, right guard, center, left power forward, right power forward)? ii) How many ways can he choose 5 players if the tallest player must be the center? KEEP TOGETHER Permutations Problems with Conditions/Cases 7 A) How many arrangements can be made from the word LOCKERS if L and S must be kept together? B) How many ways can you stand Amanda, Zachary, Garrett, Nicole and Mitchell in a line if: i) there are no restrictions ii) Garrett and Nicole must be side by side. (Treat as One) iii) Amanda and Mitchell cannot stand side by side. (Treat Separately)
AT MOST/AT LEAST Consider multiple cases then add the results 8 To open a lock on a storage locker door pass codes must be comprised from at least a 4, 5, or 6 digit pass code from the digits 0 to 9. If repetition is not permitted, determine the total number of pass codes possible. (3 cases) 9 An alarm code must contain at least 3 letters up to a maximum of 5 letters along with 3 digits selected from the numbers 0 to 9. A) Determine the total amount of codes possible if repetition is possible amongst the letters only. (3 cases) B) If repetition is not permitted whatsoever. 10 Determine all the arrangements (of any number of letters 1, 2, 3, or 4) of the word TASK?
Type 3 Arrangements when items are Identical. n! abc!!! 11 How many ways can you arrange the word BANANAS? 12 Show that there are only 120 distinct 5 letter arrangements of the word GREAT but only 60 of the word GREET. 13 A clerk has 3 different best selling novels. He has 6 copies of each. A) How many different ways can you arrange theses books on the shelf if the copies of the same novel must be kept together? (Items together treat as one of each) See #7 page 105 textbook. B) How many different ways can all the books be arranged? 14 How many routes are there from A to B if you travel only South or East? 15 How many ways can you arrange the letters in CANADA if C must be first? 16 Compute by showing all workings: Combinations nc r Or n r A) 15 C 6 B) 9 3 C) 30C 29
17 Solve for n: 1C1 40 n Combination Problems with Conditions/Cases 18 The student council has decided to form a sub-committee of 6 members to plan graduation decoration. There are no positions on this committee. There are a total of 13 student council members: 6 male, 7 female. A) How many ways can the committee consist of EXACTLY 4 males? B) How many ways can the committee consists of at least 4 females? (3 cases here!) C) How many committees can be formed with no males?