Unit 6 Permutations and Combinations 6 1 Fundamental

Unit #6 Permutations and Combinations 6. 1 Fundamental Counting Principle The fundamental counting principal is a mathematical method of finding the number of ways an outcome can be performed. Bert Church High School has 7 doors to enter/exit the building. How many ways can a student enter, then exit the building if they do not use the same door twice? Enter Exit Total Results 1 2, 3, 4, 5, 6, or 7 6 2 1, 3, 4, 5, 6, or 7 6 3 1, 2, 4, 5, 6 or 7 6 4 1, 2, 3, 5, 6 or 7 6 etc Ways to enter Ways to exit

6. 1 Fundamental Counting Principle cont. An outcome is made up of events a, b, c, d, etc. The number of ways this outcome can be performed is given by; EXAMPLE #1 Class example #1, pg. 373 A toy manufacturer makes wooden toys in three parts. The top part is either red, white or blue. The middle part is orange or black. The bottom part is yellow, green, pink or purple. a. How many different coloured toys are possible using a tree diagram? b. How many different coloured toys are possible using the fundamental counting principle?

6. 1 Fundamental Counting Principle cont. EXAMPLE #2 Class example #3, pg. 374 A Math 30 quiz consists of eight multiple choice questions. Each question has four choices, A, B, C, and D. How many different sets of answers are possible? EXAMPLE #3 Class example #5, pg. 374 Car number plates in an African country consist of a letter other than I or O, followed by three digits, the first of which cannot be zero, followed by any two letters which are not repeated. How many different car number plates can be produced?

6. 1 Fundamental Counting Principle cont. EXAMPLE #4 Class example #6, pg 375 Consider the digits 2, 3, 5, 6, 7 and 9. a) If repetitions are not permitted, how many 3 -digit numbers can be formed? b) How many of these are less than 400? Even? Odd? Multiples of 5?

6. 2 Permutations are an arrangement where order is important. Subset Permutations A subset permutation consists of n possible events where only r are chosen. 8 people enter a relay race. How many ways can they finish 1 st, 2 nd and 3 rd? This example illustrates that if there are n items, and we choose r of them, we can arrange those items; ways

6. 2 Permutations cont. Full set Permutations A full set permutation consists of n possible events where all n are chosen. 8 people enter a relay race. How many ways can they finish? This is more commonly calculated as 8! MATH - PRB This example illustrates that if there are n items, and we choose n of them, we can arrange those items;

6. 2 Permutations cont. Calculating a permutation using a formula 8 people enter a relay race. How many ways can they finish 1 st, 2 nd and 3 rd?

6. 2 Permutations cont. EXAMPLE #1 Class example #1, pg 377. Find the value of EXAMPLE #2 Class example #2, pg 377 Simplify

6. 2 Permutations cont. EXAMPLE #3 Class example #5, pg. 381 In a South American country, vehicle license plates consist of any 2 different letters followed by 4 different digits. Find how many different license plates are possible by; a) Fundamental counting principle b) permutations EXAMPLE #4 Class example #6, pg. 382 Solve

6. 3 Permutations with Restrictions/Repetition How many different arrangements of the word CANADA are possible? This would only be true if all the letters in the word CANADA were unique (i. e. no repetition of the letter A). Examine the following arrangements; AAACND AAACND The calculation considers each of these to be different, based on the assumption that the A’s are unique. The reality is that these 6 arrangements are not different, and should only be counted once. Arrangements of the word CANADA …

6. 3 Permutations with Restrictions/Repetition cont. General Rule for Repetition The number of arrangements of n items, where p are of one kind, q are of a second kind, r are of a third kind, etc, is given by; EXAMPLE #1 Class example #1, pg. 385 How many ways can all the letters of the word ORANGES be arranged if; a) There are no restrictions. b) The first letter must be an N. c) The vowels must be together.

6. 3 Permutations with Restrictions/Repetition cont. EXAMPLE #2 Class example #3, pg. 385 Find the number of permutations of the letters in the word KITCHEN if; a) The letters K, C and N must be together but not necessarily in that order. b) The vowels must not be together. EXAMPLE #3 Class example #4, pg. 385 In how many ways can 3 girls and 4 boys be arranged in a row if no two people of the same gender can sit together?

6. 3 Permutations with Restrictions/Repetition cont. EXAMPLE #4 Class example #7, pg. 387 Find the number of permutations of the letters of the word; a) VANCOUVER b) MATHEMATICAL EXAMPLE #5 Class example #8, pg. 387 How many arrangements of the word POPPIES can be made under each condition; a) b) c) d) e) Without restrictions. Each arrangement begins with a P. The first two letters are P. All the P’s are together. The arrangement begins with exactly one P.

6. 4 Combinations are arrangements where order is not important. In a class of 15 students, 3 will be selected to win one of 3 prizes, each $50. 00. How many groups of 3 students can be selected? Examine the following groups of three students, D, F and I. DFI DIF FDI FID IDF IFD will count these 6 times. However, this is the same group of three students, and should only be counted once (they don’t care about the order because all prizes are the same).

6. 4 Combinations cont. Calculating a combination using a formula 15 items choose 3, order not important. MATH - PRB

6. 4 Combinations cont. EXAMPLE #1 Class example #2, pg. 395 To win LOTTO 649, a person must correctly choose six numbers from 1 -49. Jasper , wanting to play LOTTO 649, began to wonder how many numbers he could make up. How many choices would Jasper have to make to ensure he had the six winning numbers? EXAMPLE #2 Class example #3, pg. 395 The Athletic Council decides to form a sub-committee of seven council members to look at how funds raised should be spent on sports activities in the school. There a total of 15 athletic council members, 9 makes and 6 females. The subcommittee must consist of exactly 3 females; a) b) c) d) How many ways can the females be chosen? How many ways can the males be chosen? In how many ways can the subcommittee be chosen if Bruce, the football coach, must be included?

6. 4 Combinations cont. EXAMPLE #3 Class example #1, pg. 399 The Student Council decides to form a subcommittee of five council members to look at how funds raised should be spent on the students of the school. There a total of 11 student council members, 5 males and 6 females. How many different ways can the subcommittee consist of; a) Exactly 3 females? b) At least 3 females? c) At least one male? EXAMPLE #4 Class example #5, pg. 401 During a Pee Wee hockey tryout, all the players met on the ice after the last practice and shook hands with each other. How many players attended the trout if there were 300 handshakes in all?

6. 4 Combinations cont. EXAMPLE #5 Class example #6, pg. 402 A polygon has 65 diagonals. How many sides does it have?

6. 5 Probability using Permutations and Combinations Probability is based on the calculation of a ratio. The probability of event E is; Counting methods (fundamental counting principle, permutations and combinations) allow us to determine both the favourable and possible outcomes. EXAMPLE #1 Class example #1, pg 419. The word COUNTED has been spelled using Scrabble tiles. Two tiles are randomly chosen at a tie and placed in the order which they were chosen. Determine the probability that the tiles are: a) CO b) Both vowels

6. 5 Probability using Permutations and Combinations cont. EXAMPLE #2 Class example #2, pg. 419 The Athletic Council decides to form a subcommittee of seven council members to look at how funds raised should be spent on sports activities at the school. There a total of 15 athletic council members, 9 males and 6 females. What is the probability that the subcommittee will consist of exactly 3 females? EXAMPLE #3 Class example #5, pg. 420 In a card game, you are dealt 5 cards from a pack of 52 shuffled cards. When you look at your 5 cards, what is the probability, expressed in combination notation, you have; a) b) c) d) Four aces? Four tens and an ace? 10, J, Q, K and ace? At least one Jack?

6. 6 Pascal’s Triangle and Pathways Mathematician Blaise Pascal discovered a number pattern in the seventeen century that has many applications to mathematical problems today. The pattern is; Can you find the next two rows? Add adjacent numbers in the previous row to find the numbers making up the next row.

6. 6 Pascal’s Triangle and Pathways cont. Pascal Triangle Properties Many interesting phenomena can be observed from the number pattern; + + + The nth row has a sum of Add these numbers … And, you get this one. This works on any diagonal.

6. 6 Pascal’s Triangle and Pathways cont. Pascal Triangle Properties and Combinations Each row of the triangle is formed by the combinations;

6. 6 Pascal’s Triangle and Pathways cont. EXAMPLE #1 Class example #1, pg 407 a) What is the sum of the numbers in the tenth row of Pascal’s Triangle? b) Omit for now. c) What is the sum of; EXAMPLE #2 Class example #1, pg. 413 A city centre has a rectangular road system with 5 streets running north to south and 6 avenues running west to east. a) Draw a grid to represent this situation. b) Sean is driving a car situated at the extreme northwest corner of the city centre. How many ways can he drive to the extreme southeast corner?

6. 6 Pascal’s Triangle and Pathways cont. EXAMPLE #3 Class example #2, pg 414 Find the number of pathways from A to B if paths must move closer to B.

6. 6 Pascal’s Triangle and Pathways cont. EXAMPLE #4 Class example #3, pg. 414 How many routes must be considered if, at each intersection the taxi must always move closer to the stadium?

6. 7 Binomial Theorem Expand each of the following; Notice; 1. 2. 3. 4. The x value in the first term is to the power of n. The x value in successive terms is reduced by 1. The y value in the first term is to the power of 0. The y value in successive terms is increased by 1 until the last term where it’s n.

6. 7 Binomial Theorem cont. The derivation of coefficients Examine The coefficients of 1, 3, 3, 1 are found in Pascal’s Triangle. Here …

6. 7 Binomial Theorem cont. Putting it all together Expand With n = 5, the final expression will contain 6 terms, so you must use the 6 th row of Pascal’s Triangle.

6. 7 Binomial Theorem cont. EXAMPLE #1 Class example #2, pg. 409 Find the fifth term of Find the middle term of EXAMPLE #2 Class example #4, pg. 409 One term in the expansion of EXAMPLE #3 Class example #5, pg. 409 Find the constant term of is . Find a.

6. 8 Binomial Probability Over the next few weeks, the Flames will play the Oilers 4 times. In the last 10 games they’ve played, the Flames have won 7 times. What is the probability the Flames will beat the Oilers in 3 of their next 4 games? Based on what we know, each time they play, the probability of a Flames win is; So, a four game stretch might look like; WWWL (W is a win, L is a loss). Win Loss

6. 8 Binomial Probability cont. What is incorrect about Why should we be restricted to the first 3 games being wins? Is this the only arrangement? Arrangements of 3 W’s and 1 L or

6. 8 Binomial Probability cont. Probability of the Flames beating the Oilers in 3 of the next 4 games? Arrangements possible for wins/losses. Wins Losses Total possible outcome for the Flames and Oilers playing 4 games? Flames win all Flames win 3 Flames win 2 Flames win 1 Flames don’t win

6. 8 Binomial Probability cont. EXAMPLE #1 Class example #2, pg 435 Five dice are rolled. What is the probability of rolling a) Exactly three twos? b) At least three twos?