Counting Methods. Mathematics 3201

Transcription

1 Mathematics 3201 Unit COUNTING PRINCIPLES Goal: Determine the Fundamental Counting Principle and use it to solve problems. Example 1: Hannah plays on her school soccer team. The soccer uniform has: three different sweaters: red, white, and black, and three different shorts: red, white, and black. How many different variations of the soccer uniform can the coach choose from for each game? Make a tree diagram. Try: A toy manufacturer makes a wooden toy in three parts. Determine how many different coloured toys can be produced? Make a tree diagram. Part 1: The top part may be coloured red or blue Part 2: The middle part may be orange, white, or black Part 3: The bottom part may be yellow or green Fundamental Counting Principle If there are a ways to perform one task and b ways to perform another, then there are a b ways of performing both. Consider a task made up of several stages. The fundamental counting principle states that if the number of choices for the first stage is a, the number of choices for the second stage is b, the number of choices for the third stage is c, etc.. then the number of ways in which the task can be completed is a b c... 1

2 Mathematics 3201 Unit 2 Solve a Counting Problem by Extending the Fundamental Counting Principle Example 2: A luggage lock opens with the correct three-digit code. Each wheel rotates through the digits 0 to 9. a. How many different three-digit codes are possible? b. Suppose each digit can be used only once in a code. How many different codes are possible when repetition is not allowed? Try: A vehicle license plate consists of 3 letters followed by 3 digits. How many license plates are possible if: a. there are no restrictions on the letters or digits used? b. no letter may be repeated? c. the first digit cannot be zero and no digits can be repeated? Example 3: Solving a counting problem when the Fundamental Counting Principle does not apply A standard deck of cards contains 52 cards as shown. Count the number of possibilities of drawing a single card and getting: a. either a red face card or an ace b. either a club or a two 2

3 Mathematics 3201 Unit Counting Principles Example 4 Recap Example 1: Hannah plays on her school soccer team. The soccer uniform has: Three different sweaters: red, white, and black, and Three different shorts: red, white, and black. How many different variations of the soccer uniform can the coach choose from for each game? Method 1: Use a tree diagram Method 2: Use the Fundamental Counting Principle Number of ways to choose a sweater Number of ways to choose shorts Fundamental Counting Principle If there are a ways to perform one task and b ways to perform another, then there are ways of performing both. Number of uniform variations = x = There are different variations of the soccer uniform to choose from. Example 5: A bike lock opens with the correct four-digit code set by rotating five wheels through the digits 0 to 9. a) How many different four-digit codes are possible? # of ways to choose 1st digit # of ways to choose 2nd digit # of ways to choose 3rd digit # of ways to choose 4th digit Number of different codes = x x x = There are different four-digit codes. b) Suppose each digit can be used only once in a code. How many different codes are possible when repetition is not allowed? Number of different codes = x x x = There are different four-digit codes when the digits cannot repeat. 3

4 Mathematics 3201 Unit 2 The Fundamental Counting Principle applies when tasks are related by the work AND If tasks are related by the work OR: If the tasks are mutually exclusive, they involve two disjoint sets A and B: If the task s are not mutually exclusive, they involve two sets that are not disjoint, C and D: The Principle of Inclusion and Exclusion must be used to avoid counting elements in the intersection of the two sets more than once. Example 7: A standard deck of cards contains 52 cards. Count the number of possibilities of drawing a single card and getting: a) either a black face card or an ace There are ways to draw a single card and get either a black face card or an ace. b) either a red card or a 10 There are ways to draw a single card and get either a red card or a 10. 4

5 Mathematics 3201 Unit INTRODUCING PERMUTATIONS AND FACTORIAL NOTATION Permutation is an arrangement of distinguishable objects in a definite order. For example, the objects a and b have two permutations, ab and ba. Solve a Counting Problem Where Order Matters Example 1: Determine the number of arrangements that 4 children can form while lining up to washroom. Example 2: When you press the shuffle button on an i-pod, it plays a list of the songs (all songs will be played only once). If the i-pod has 6 songs on it, how many playlists of the songs are possible? Try: How many different ways can 5 different books, Math, Chemistry, Physics, English and Biology be arranged on a shelf? Factorial Notation! 1! = 2! = 3! = 4! = 5! = A concise representation of the product of consecutive descending natural numbers: n! = (n +1)! = (n 1)! = 6! = 5

6 Mathematics 3201 Unit 2 In the expression n!, the variable n is defined only for values that belong to the set of whole numbers; that is, n { 0, 1, 2, 3,...}. Please note that 0! is defined to be 1. Evaluate Numerical Expressions Involving Factorial Notation Example 3: Evaluate the following. 10! 12! 9!3! Simplify an Algebraic Expression Involving Factorial Notation Example 4: Simplify each expression, where n N ! +1! 1! Example 5: Write each expression without using the factorial symbol. + 2!! 3!! Try: Calculate the value of: 43! 40! 37! 33!4! 6

7 Mathematics 3201 Unit 2 Solve an Equation Involving Factorial Notation Example 6: Solve for n.! =90, where!!! = 126, where 7

8 Mathematics 3201 Unit Introducing Permutations and Factorial Notation Factorial Notation A concise representation of the product of consecutive natural numbers: 1! = 2! = 3! = 8! = n! = Permutation An arrangement of distinguishable objects in a definite. For example, the objects a and b have two permutations: and Example 7: Determine the number of arrangements that six children can form while lining up to drink. There are six children in the lineup, so there are six possible positions: Let L represent the total number of permutations: L = L = Factorial Notation There are permutations of the six children at the fountain. Example 8: British Columbia licence plates for passenger cars have 3 numbers followed by 3 letters. The letters I, O, Q, U, Y, Z are not used. How many possible permutations are there of licence plates? 8

9 Mathematics 3201 Unit Introducing Permutations and Factorial Notation Example 9: Evaluate the following. a) 10! b)!!! Example 10: Simplify, where a) 3 2! b)!! 0! is defined to be equal to Restrictions on n if n! is defined: For example: State the values of n for which each expression is defined, where. a) 3! b)!! 9

10 Mathematics 3201 Unit 2 Example 11: Solve! 6, where I.! Example 12: Solve!! 90, where I.

11 Mathematics 3201 Unit Permutations When All Objects Are Distinguishable The number of permutations from a set of n different object, where r of them are used in each arrangement, can be calculated using the formula: np r!, where 0! Example 1: Matt has downloaded 10 new songs from an online music store. He wants to create a playlist using 6 of these songs arranged in any order. How many different 6-song playlists can be created from his new downloaded songs? Method 1: Use the n P r formula: Method 2: Use the Fundamental Counting Principle Example 2: Tania needs to create a password for a social networking website she registered with. The password can use any digits from 0 to 9 and/or any letters of the alphabet. The password is case sensitive, so she can use both lower- and upper-case letters. A password must be at least 5 characters to a maximum or 7 characters, and each character can be used only once in the password. How many different passwords are possible? 11

12 Mathematics 3201 Unit 2 Example 3: At a used car lot, seven different car models are to be parked close to the street for easy viewing. a) The three red cars must be parked so that there is a red car at each end and the third red car is exactly in the middle. How many ways can the seven cars be parked? b) The three red cars must be parked side by side. How many ways can the seven cars be parked? Example 4: A social insurance number (SIN) in Canada consists of a nine-digit number that uses the digits 0 to 9. If there are no restrictions on the digits selected for each position in the number, how many SINs can be created if each digit can be repeated? How many SINs can be created if no repetition is allowed? In reality, the Canadian government does not use 0, 8, or 9 as the first digit when assigning SINs to citizens and permanent residents, and repetition of digits is allowed. How many nine-digit SINs do not start with 0, 8, or 9? 12

13 Mathematics 3201 Unit Permutations of Distinct Objects If you are selecting items from one group, and then arranging them in some kind of order, the number of possibilities are called. You can calculate them using the Fundamental Counting Principle Example 1. There are 7 books in a box. How many ways can you line up 5 of them on a shelf? But sometimes there are too many items in the group or spots to fill for this to be practical Example 2. There are 72 characters you could use for a password. You are not allowed to repeat characters. How many different 8-character passwords are possible? In these cases, you can use the permutation formula: Example 3 Example 4 P = P = n P r = n! ( n r)! Other examples: 13P P = P = P =

14 Mathematics 3201 Unit 2 Permutation Problems with Cases Example 5: Determine the number of possible accounts with 4 character passwords or 5 character passwords. No repeats allowed. Given that there are 72 possible characters. Permutation Problems with Restrictions Example 6: How many possible arrangements for a photo. Given that 20 people, which includes 12 adults and 8 kids.the kids must be placed in the front row. The adults will be placed in the back row. Example 7: Solving a Permutation Equation P n 2 = 42 8 P r =

15 Mathematics 3201 Unit 2 Investigate 2.4 Permutations When Objects are Identical 1. The permutations of the 4 different letters A, B, E, and R are: ABEF ABFE AEBF AFBE AEFB AFEB BAEF BAFE EABF FABE EAFB FAEB BEAF BFAE EFAB FEAB EBAF FBAE BEFA BFEA EFBA FEBA EBFA FBEA How many permutations are there? 2. a) What happens if two of the letters are the same? Investigate this by converting each F to an E in the list below. Then count the number of permutations of the letters A, B, E, and E. ABEF ABFE AEBF AFBE AEFB AFEB BAEF BAFE EABF FABE EAFB FAEB BEAF BFAE EFAB FEAB EBAF FBAE BEFA BFEA EFBA FEBA EBFA FBEA There are permutations of the letters A, B, E, and E. b) How does this number compare with Exercise 1? 3. What happens if three of the letters are the same? Investigate this by converting each F and E to a B. Then count the number of permutations of the letters A, B, B, and B. ABEF ABFE AEBF AFBE AEFB AFEB BAEF BAFE EABF FABE EAFB FAEB BEAF BFAE EFAB FEAB EBAF FBAE BEFA BFEA EFBA FEBA There are permutations of the letters A, B, B, and B. EBFA FBEA b) How does this number compare with Exercise 1? 17

16 Mathematics 3201 Unit 2 4. Generalize the pattern from the investigation on the previous page to determine the number of permutations of a) A, B, C, D, D b) A, B, D, D, D c) A, D, D, D, D d) A, B, B, C, C e) A, A, A, B, B Generalization The number of permutations of n objects, where a are identical, another b are identical, another c are identical, and so on, is: Example 1: Determine the number of permutations of all the letters in the following the words. a) STATISTICIAN b) CANADA Example 2: How many ways can the letters of the word CANADA be arranged if the first letter must be N and the last letter must be C? 18

17 Mathematics 3201 Unit 2 Example 3: Julie s home is three blocks north and five blocks west of her school. How many routes can Julie take from home to school if she always travels either south or east? Method 1: Using Permutations Possible Routes: Method 2: Using a diagram 19

18 Mathematics 3201 Unit PERMUTATION WHEN OBJECTS ARE IDENTICAL Goal: Determine the number of permutations when some objects are identical. Example 4: Three cans are to be put on a shelf. a. List all permutations. b. If the Red Bull is replaced by another Coca-Cola, list all permutations. The number of permutations of n objects, where a are identical, another b are identical, another c are identical, and! so on, is!!!. Example 5: Beck bought a carton containing 6 mini boxes of cereal. There are 3 boxes of Cheerios, 2 boxes of Fruit Loops, and 1 box of Mini-Wheats. Over a six day period, Beck plans to eat the contents of one box of cereal each morning. How many different orders are possible? 20

19 Mathematics 3201 Unit 2 Try: Naval signals are made by arranging coloured flags in a vertical line and the flags are then read from top to bottom. How many signals using six flags can be made if you have: a. 3 red, 1 green, and 2 blue flags b. 2 red, 2 green, and 2 blue flags Example 6: Determine the number of permutations of all the letters in each of the following words. a. OGOPOGO b. STATISTICIAN Solve a Conditional Permutation Problem Involving Identical Objects Example 7: How many ways can the letters of word CANADA be arranged, if the first letter must be N and the last letter must be C? Try: Tina is playing with a tub of building blocks. The tub contains 3 red blocks, 5 blue blocks, 2 yellow blocks, and 4 green blocks. How many different ways can Tina stack the block in a single tower, if there must be a yellow blocks at the bottom of the tower and a yellow block at the top. 21

20 Mathematics 3201 Unit 2 Example 8: A supervisor of the city bus department is determining how many routes there are from the bus station to the concert hall. Determine the number of routes possible if the bus must always move closer to the concert hall. Example 9: A taxi company is trying to find the quickest route during rush hour traffic from the train station to the football stadium. How many different routes must be considered if at each intersection the taxi must always move closer to the football stadium? 23

21 Mathematics 3201 Unit 2 2.4: Permutations of Identical Objects How many ways could you arrange 8 Scrabble tiles if they were all different? But what if some of the tiles are identical? How many arrangements are unique? Try it with three tiles and no repeats: apt, atp, pat, pta, tap, tpa Try it with three tiles and 2 repeats: att, att, tat, tta, tat, tta Formula: n! P = So, distinct re-arrangements of SPARTANS = abc!!!... Try these examples: How many distinct arrangements can be made from the letters INVITATION? A string of 12 lights can be either on or off to send a message. How many different messages can be sent if exactly 5 of the lights are on? 24

22 Mathematics 3201 Unit 2 Pathway Problems and Pascal s Triangle 25

23 Mathematics 3201 Unit & 2.6 Combinations Investigate: 1. If 5 sprinters compete in a race, how many different ways can the medals for first, second and third place, be awarded? Does order matter here? This is an example of a permutation of objects, taken at a time. 2. If 5 sprinters complete in a race and the fastest 3 qualify for the relay team, how many different relay teams can be formed? Visualize the 5 sprinters below. Since 3 will qualify for the relay team and 2 will not, consider the number of ways of arranging 3 Y s and 2 N s. A grouping of objects where. Does the order of finish matter here? Combinations This is an example of a combination of objects, taken at a time Ex: The two objects a and b have on combination because is the same as The number of combinations from a set of n different objects, where only r of them are used in each combination, can be denoted by or (read n choose r ), and is calculated using the formula: Example 2: A group of 7 people consists of 3 males and 4 females. a) How many different committees of 3 people can be formed from 7 people? b) How many different committees of 3 people can be formed if the first person selected serves as the chairperson, the second as the treasurer, and the third as the secretary? 26

24 Mathematics 3201 Unit 2 c) How many different committees of 3 people can be formed with 1 male and 2 females? Think: you must choose 1 male out of the group of 3 males and 2 females out of the group of 4 females d) How many different committees of 3 people can be formed with at least one male on the committee? Notes: The formula for n C r is the formula for n P r divided by. Dividing by eliminates the counting of the same combination of r objects arranged When solving problems involving combinations, it may also be necessary to use the Sometimes combination problems can be solved using direct reasoning. This occurs when there are conditions involved. To do this, follow the steps below: 1. Consider only the cases that reflect the 2. Determine the of combinations for each case. 3. the results of step 2 to determine the total number of combinations. Sometimes combination problems that have conditions can be solved using indirect reasoning. To do this, follow these steps: 1. Determine the of combinations without any conditions. 2. Consider only the cases that meet the conditions. 3. Determine the number of combinations for each case identified in step the results of step 3 from step 1. 27

25 Mathematics 3201 Unit & EXPLORING COMBINATIONS Goal: Solve problems involving combinations. Example 3: Calculating Combinations If 5 sprinters compete in a race final, how many different ways can the medals for first, second, and third place be awarded? If 5 sprinters compete in a qualifying heat of a race, how many different ways can the sprinters qualify? A permutation is an arrangement of elements in which the order of the arrangement is taken into account. A combination is a selection of element in which the order of selection is NOT taken into account. Example 4: Solving a Simple Combination Problem Three students from a class of 10 are to be chosen to go on a school trip. In how many ways can they be selected? Combination of n different objects taken r at a time is: n C r n n! = = r r!( n r)! There are 16 students in a class. Determine the number of ways in which four students can be chosen to complete a survey. 28

26 Mathematics 3201 Unit 2 Example 5: Solving a Combination Problem Using the Fundamental Counting Principle The Athletic Council decides to form a sub-committee of 7 council members to look at how funds raised should be spent on sport activities in the school. There are a total of 15 athletic council members, 9 males and 6 females. The sub-committee must consist of exactly 3 females. Determine the number of ways of selecting the sub-committee. A basketball coach has 5 guards and 7 forwards on his basketball team. In how many different ways can he select a starting team of two guards and three forwards? Example 6: Solving a Combination Problem by Considering Cases A planning committee is to be formed for a school-wide Earth Day program. There are 13 volunteers: 8 teachers and 5 students. How many ways can the principal choose a 4-person committee that has at least 1 teacher? An all-night showing at a movie theatre is to consist of five movies. There are fourteen different movies available, ten disaster movies and four horror movies. How many possible schedules include at least four disaster movies? 29

27 Mathematics 3201 Unit 2 Practice being able to translate the formula into factorials quickly: C C C C4 Practice recognizing when the order matters or not Say whether each of these involve a permutation (P) or a combination (C) a) the number of ways that 3 horses out of 8 could end up first, second and third in a race b) the number of different hands of 5 face cards you could draw from a deck of 52 cards c) the number of ways a team could win 2 or 3 games out of the next 5 d) the number of different messages you could send with three coloured flags out of a set of 10 flags e) the number of ways you can rearrange the letters in the word BASEBALL f) the number of ways to pick a group of 2 boys and 3 girls from a class of 14 boys and 15 girls g) the number of possible winning numbers in a lottery where 6 numbers are chosen out of a total of 49 31

28 Mathematics 3201 Unit Solving Counting Problems Example #1: Mrs. Vos and some of her favourite students are having a group photograph taken. There are three boys and five girls. The photographer wants the boys to sit together and the girls to sit together for one of the poses. How many was can the students and teacher sit in a row of nine chairs for this pose? Example #2: A standard deck of 52 playing cards consists of 4 suits (spades, hearts, diamonds, and clubs) of 13 cards each. a) How many different 5-card hands can be formed? b) How many different 5-card hands can be formed that consist of all hearts? c) How many different 5-card hands can be formed that consist of all face cards? d) How many different 5-card hands can be formed that consist of 3 hearts and 2 spades? 34

29 Mathematics 3201 Unit 2 e) How many different 5-card hands can be formed that consist of at least 3 hearts? f) How many different 5-card hands can be formed that consist of at most 1 black card? When solving counting problems, you need to determine if plays a role in the situation. Once this is established, you can use the appropriate permutation or combination formula. You can also use these strategies: Look for. Consider these first as you develop your solution. If there is a repetition of r of the n objects to be eliminated, it is usually done by If a problem involves multiple tasks that are connected by the word, then the Fundamental Counting Principle can be applied: the number of ways that each task can occur. If a problem involves multiple tasks that are connected by the word, the Fundamental Counting Principle apply; the number of ways that each task can occur. This typically is found in counting problems that involve. 35

30 Mathematics 3201 Unit 2 Chapter 4 Review!!!! n P r = n C r =!!!!! 1. Solve for n. a)! 12 b)!!! A computer store sells 5 different desktop computers, 4 different monitors, 6 different printers, and 3 different software packages. How many different computer systems can the employees build for their customers? 3. From a standard deck of cards, how many possibilities are there for drawing a) a two or a face card? b) a spade or a queen? 4. How many arrangements are possible using all the letters in a) WHISTLER b) REARRANGE 5. How many routes are there from A to B in each map, if you only travel south and west.? a) b) 32

31 Mathematics 3201 Unit 2 6.Gareth works at an electronics store. He has 6 copies of 3 different new DVD releases to put on a shelf. a) In how many different ways can he arrange the DVDs on the shelf? b) In how many different ways can the DVDs be arranged if the copies of each one must be grouped together? 7. At a car lot, six different coloured cars are to be parked close to the street. The cars are blue, red, white, black, silver, and green. a) How many ways can the cars be parked? b) How many ways can the cars be parked such that the white and black cars are always next to each other. 8. Twelve students are running for student council. How many ways can a student council of three be chosen? 9. Twelve students are running for president, treasurer and secretary of the class. How many ways can the three positions be filled? 10. There are 14 women and 8 men who audition for an improve team. a) How many different combinations of people could be chosen for the team if there are 3 women and 2 men chosen? b) How many different combinations of 5 people could be chosen if at least 3 men are chosen. 33