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n! n(n 1)(n 2)...3 2 1 n Pr n! (n r )! In the expansion of where n N and 0! 1. nCr ( x y)n , n! (n r )!r ! the general term is or nr tk 1 n Ck x nk y k Math 30-1 AP Mrs. D. Atkinson 1 2 Math 30-1 Unit: Permutations and Combinations Topic: The Fundamental Counting Principle Objective: The students will apply the fundamental counting principle and factorial notation to various problems. Example In how many ways can the letters ABC be arranged? Example A toy manufacturer makes a wooden toy in three parts: Part 1: the top part may be colored red, white or blue. Part 2: the middle part may be colored orange or black Part 3: the bottom part may be yellow, green, pink or purple Determine how many different colored toys can be produced. The total number of choices is __________. This can be obtained by counting the choices summarized at the side OR by using a method called the Fundamental Counting Principle (FCP). The FCP multiplies individual choices together to obtain the final number of choices. In the above example, you would multiply 3 by 2 by 4 to get a final number of _________. NOTE: One of the key things to realize from this point on is that the English word AND implies multiplication when translated to mathematics and the English word OR implies addition. (Ds do not go together) 3 Example In how many ways can the letters of the word FUNCTION be arranged? Fundamental Counting Principle (FCP) Example A new car is available in 3 models, 6 colors, 2 transmissions and 4 different option packages. How many versions of the car can be created? Example A math quiz consists of eight multiple choice questions. Each question has four choices A, B, C or D. How many different sets of answers are possible? Example In how many ways can 5 paintings be arranged on a wall? Example In how many ways can the letters of the word BOLAGNY be arranged? (and yes, I realize this is not how to spell the word “bologna” ) 4 Example How many different routes are there to travel from Edmonton to Lethbridge through Red Deer and Calgary as shown on the map? E R C L Example There are two routes from Pitland to Queensville, three routes from Quennsville to St. Luke’s, three routes from Pitland to Rutherford, and one route from Rutherford to St. Luke’s. a. How many routes are there from Pitland to St. Luke’s passing through Queensville? b. How many routes are there from Pitland to St. Luke’s passing through Rutherford? c. How many routes are there from Pitland to St. Luke’s? Example In how many ways can the letters of the alphabet be arranged? Hm… this might take a while 5 Factorial Notation and the Permutation Formula Part A: Factorial Notation and Definition n! This notation means that you are multiplying in descending order starting at “n” and ending at 1. Mathematically, this can be given as: GENERAL CASE: n! n n 1 n 2 ...3 2 1 where n W Why is n W ? ______________________________________________________________________________ EXAMPLE: 6! 6 5 4 3 2 1 You can use a calculator keying sequence short cut to 6 5 4 3 2 1 . It is found under Math>Prb>4! Example Evaluate: a. 11! Example Write as a single factorial. a. 27 x 26 x 25 x 24 x 23! b. 100! 98! c. 7 ! d. 0! b. x 3 x 4 x 5 ... 3 2 1 Example Write 10 9 8 7 as a quotient of factorials. 6 Example Simplify: a. c. n 1! n 1! n 2 ! n ! b. n 3 ! n 1! d. n ! n n 1 Example n! 42 Solve n 2 ! 7 Part B: Connection to FCP F.C.P. (the Fundamental Counting Principle) finds the total number of choices for a certain event by multiplying the individual choices together. It is used to find the arrangement of a set of objects where order is important (this is called a permutation). EXAMPLE: Arranging three from a choice of seven different toys in a row. Why is this called a permutation? In other words, why does order matter? NOTE: The word “arrange” usually implies a permutation. NOTE: If, instead of lining these toys up, we picked out three to throw in the diaper bag, then order wouldn’t matter – that would be a combination. Say you have seven different toys and you arrange all seven in a row. Consider this example two ways below. FCP: Factorial: This is the number of permutations of “n” different objects taken all at time – n! Say you have seven different toys and you arrange three of the seven in a row. Consider this example two ways below. FCP: Factorial: This is the number of permutations of “n” different objects taken “r” at time – n! Note A permutation is an arrangement of all or part of a set of objects. The order of arrangement is n! important. If n objects are arranged r at a time, then there are n Pr arrangements. n r ! 8 NOTE ABOUT 0! “n” different objects taken “n” at a time, we know is n! n! Substitute that into the formula: n Pr = n r ! For this to equal to n! , 0! must equal 1. Example Use the formula to evaluate P. 11 3 Example How may 4 letter arrangements can be made using the letters of the word a. FACTOR b. HYPERBOLA Example In a region, vehicle license plates consist of 2 different letters followed by 4 different digits. If the letters I, O, Y and Z are not used, determine how many different license plates are possible by a. Fundamental counting principle b. permutations Example A super duper athlete has 10 different trophies. Her shelf can only hold 7 of them. How many different ways can the trophies be arranged on the shelf? Example Mr. Keener Teacher has 10 multiple choice questions on Permutations. He can only put four of them on the test. How many unique tests can Mr. Math make? 9 Example We still use the FCP the majority of the time as we learned it first and sometimes it is simpler. We need to know the formula; however, for algebraic questions…. For example, solve for ”n” in the equation n P4 28 n 1 P2 STEPS: 1. Simplify with our n Pr formula 2. Cross Multiply. 3. Simplify with the factorial definition. Eliminate like factorials on each side of the equals sign when possible. 4. Solve the remaining quadratic equation and check for extraneous roots. 10 Example n! Solve 84 n2 Pn 4 Example Solve n1 P2 90 Textbook Page 524 #1-4, 6, 7, 22, 24 11 Math 30-1 Unit: Permutations and Combinations Topic: Permutations with Restrictions and Repetition Objective: The students will permutations to solve problems involving restrictions and repetition Example The telephone numbers allocated to subscribers in a rural area consist of one of the following: The digits 345 followed by any three further digits, or The digit 2 followed by one of the digits 1 to 5 followed by any three further digits How many different telephone numbers are possible? Example In an African country, license plates consists of a letter other than I or O followed by 3 digits, the first of which cannot be zero, followed by any two letters which are not repeated. How many different car license plates can be produced? Example If repetitions are allowed, how many 4 digit numbers are possible if a. there are no restrictions? b. the number cannot contain 7? c. the number must contain at least one 7? d. the number is even? 12 Example How many 4 digit numbers can be made using the digits 2, 5, 6, 7, 8 without repetition? Example How many even, four digit numbers can be made from the numerals 2, 3, 5, 8, and 9, if there are no repeated digits allowed? Example How many even, 4 digit numbers can be made from the digits 0, 2, 3, 5, 7, 8? Example How many 4 digit arrangements greater than 5000 can be made using the digits 0, 2, 5, 6, 8? Assume repetition IS allowed. Example How many numbers greater than 7000 can be made using the digits 0, 5, 6, 7, 9? Assume repetition is allowed to a maximum of 5 digits. 13 Example Determine the number of distinguishable four letter arrangements that can be formed from the word ENGLISH if; a. There are no restrictions? (always assume that letters of a word can NOT be repeated unless otherwise stated) b. The first letter must be E? c. The arrangement must contain a G? d. The first and last letters must be vowels? Example In how many ways can all the letters of the word LOGARITHM be arranged if a. there are no restrictions? b. the first letter must be M? c. the first and last letters must be vowels? d. the first two letters must be LO in any order? 14 e. the vowels must be together and the consonants must be together? f. the letters GAR must be together? g. the letters GAR must not all be together? Example How many arrangements are there of the letters of the word a. READ b. HOUSE Note So far all the examples we’ve had have had different letters but as soon as we have words with letters that repeat, we have to account for that by dividing the total number of arrangements by the number of repetitions! For example, arrangements of the letters of the word PETER would be In general, 5! 2! n! where a, b, c are “letters” that repeat. a!b!c! NOTE: If you also have restrictions, you take care of them FIRST as before. 15 Example Find the number of permutations of all the letters of the word a. ARRANGEMENTS b. MISSISSIPPI Example Brett bought a carton containing 10 mini boxes of cereal. There are 3 boxes of Corn Flakes, 2 boxes of Rice Crispies, 1 box of Coco Pops, 1 box of Shreddies, and the remainder are Raisin Bran. How many different orders are possible if on the first morning he has Raisin Bran? Example In how many ways can all the letters of the SUCCESS be arranged if a. there are no restrictions b. the first two letters must be S c. the S’s must be together Example A race at the Olympics has 8 runners. In how many orders can their countries finish if there are 2 Canadian, 1 Russian, 1 German, 1 South African and 3 American athletes? 16 Example Naval signals are made by arranging colored flags in a vertical line and the flags are then read from top to bottom. How many signals using 6 flags can be made if you have 3 red, 1 green, and 2 blue flags? Example Example A town has 6 streets running from north to south and 4 avenues running from west to east. A man wishes to drive from the extreme south-west intersection to the extreme north-east intersection, moving only north or east along one of the streets or avenues. Find the number of routes he could take. Textbook Page 524 #1,2ab,3,4abc,5,67abd,8,9,10,11,12,13,16,17,20a,22a,25,26 17 Math 30-1 Unit: Permutations and Combinations Topic: Combinations Objective: The students will apply combinations to various problems. Note When objects are selected and arranged it means that the order is important. The arrangement is a permutation. A combination is a selection of a set of objects in which the order of selection is not important, since the objects will not be arranged. Combinations 1. Discuss how throwing three of seven toys in a bag is different than lining them up… 2. For example, CFH CHF FCH FHC HCF HFC Where as these are all different permutations, they are all the same combination. This “CFH” occurs 6 (or 3! ways). Essentially, it repeats 3! times so we need to divide the 7 P3 formula by 3! n! 3. In general, that would be given as n Cr . n r !r! This formula is the ONLY way to calculate a combination (i.e. when order doesn’t matter) On your calculator, this is under Math>Prb>3. 4. The nature of this formula is that n Cr n C p if r p n . For example, 10 C6 10 C4 (note: 6 4 10 ) Check this on our calculator and then substitute both sides into our formula. In general, we say that n Cr n Cnr (note: r n r n ) Substitute this into the formula. NOTE: When you have restrictions with combinations, that alters what you are choosing from. 18 Example At a meeting, every person shakes hands with every other for a total of 300 handshakes. How many people were at the meeting? 19 - Permutation: ORDER MATTERS arranging letters of a word - arrangements of people where titles are given (i.e. president) - itineraries - Combination: ORDER DOESN’T MATTER Groups of people where titles are not given (i.e. best three go on to another level of competition or a committee where titles are not given) people sitting together or lining up for a picture card hands - handshakes - PIN number - - License plates pizza toppings (although any good cook would know otherwise) - joining dots to form figures - diagonals - - Order or winning first, second, third etc. QUESTION Permutation Combination You must use the formula! FCP Factorial Formula Restrictions - this changes what you are choosing from Restrictions - always dealt with first (even with repetitions) Repetitions - divide by the factorial of what repeats Repetitions - doesn’t happen with Combs 20 Example a. A class of 30 students is going to elect a President, Vice President and Treasurer. In how many ways can this be done? b. A class of 30 students is going to elect a committee of 3. In how many ways can this be done? Example Five students take place in a cross-country race. a. Suppose the winner of the race gets $50, the runner up gets $25 and third place gets $10. How many ways is there to award the prizes? b. In a different scenario, instead of a first, second and third place prize, three students get chosen to go to a running clinic for free. How many ways can the 5 students be chosen? c. Which scenario is a permutation (where the order in which the recipients are chosen matter) and which is a combination (where the order in which the recipients are chosen do not matter) Example Lotto 6-49 is a ticket where, out of 49 numbers, you choose 6 and hope that your numbers get picked in the weekly draw. How many different tickets are possible? 21 Example If a basketball league consists of 7 teams, how many total games must be scheduled so that each team plays every other team a. once b. three times Example Solve for x. a. 7 C2 7 Cx , x 2 b. x C3 x C7 Example A pizza joint has 9 different choices of toppings available. How many different 2-topping pizzas can be made? How many different 3-topping pizzas can be made? Example How many rectangles can be made from eight horizontal lines and three vertical lines? 22 Example A basketball coach has five guards and seven forwards on his team. In how many different ways can he select a starting team of two guards and three forwards? How many different starting teams are there if the star player, who plays guard, must be included? How many different starting teams are there if Bob and Joe, who are both guards, cannot be on the floor together? “AT LEAST”, “AT MOST”, and Which Method is Best Twelve adults and eighteen kids are on the Prom Program Planning Committee. A smaller group of them, (five people only), needs to decide on the themes. How many different ways can this smaller group (i.e. sub-committee) consist of: a) @ least four adults Method One Method Two - COMPLEMENT Figure out each situation separately Subtract the cases that aren’t “At least four”, in this case, means ____________ included from the total number 23 b) @ least one adult Method One Method Two - COMPLEMENT Figure out each situation separately Subtract the cases that aren’t “At least one”, in this case, means ____________ included from the total number c) @ most two adults Method One Method Two - COMPLEMENT Figure out each situation separately Subtract the cases that aren’t “At most two”, in this case, means ____________ included from the total number d) NOTE: Sometimes one method is more efficient than another. It is best to know how to do both. 24 Example A committee of 5 people is going to be selected from a group which contains 8 men and 7 women. How many committees are possible with . . . a. no restrictions? b. exactly 3 women? c. no men? d. at least 3 women? e. at least 1 man? f. at most 1 man? 25 Example Questions: Solve for “n” algebraically in each of the following: 1. n C1 15 C2 6 3. 4 n C2 2 n C1 2. n 4. n C2 10 Answers: 1. 2. 3. 4. 15 4 2 5 26 Decks of Cards 52 cards in a deck 4 suits – hearts (red), diamonds (red), spades (black), clubs (black) In each suit there are 3 different face cards (jacks, queens and kings) There are 13 cards in each suit. Example Using a standard deck of 52 cards, how many different five card hands are possible with a. no restrictions? b. all hearts? c. all the same suit? d. 4 queens? e. exactly 2 kings? f. at least 2 kings? g. at most 1 ace? Textbook Page 534 #1 – 20 27 Perms and Combs Problem Solving Examples 28 Example Example Assignment Page 534 #1,2,3ac,4ab,6ac,7,10,11,14a,15ab,17,18,19,20 29 Math 30-1 Unit: Permutations and Combinations Topic: Binomial Theorem Objective: The students will use the binomial theorem to expand binomials. Note The binomial theorem provides us with a procedure for expanding binomials - a binomial is ________________________________________________________ to expand a binomial means to raise the binomial to an exponent in the set of whole numbers Example Expand: x y 0 x y 1 x y 2 x y 3 x y 4 x 4 4 x 3 y 6 x 2 y 2 4 xy 3 y 4 x y 5 x 5 5 x 4 y 10 x 3 y 2 10 x 2 y 3 5 xy 4 y 5 1. What is happening to the first term (i.e. the “x” term) as we move to the right? _____________________________________________________________________ 2. What is happening to the second term (i.e. the “y” term) as we move to the right? _____________________________________________________________________ 3. How many terms does each expansion have? ____________________________ 4. How many terms does an expansion with exponent “n” have? ________________ 5. What is the degree of each term in the expansion? _________________________ 30 Binomial Expansion – Where Pascal’s Triangle Comes From 1. Fill in Pascal’s triangle into the space below 2. Where does this pattern, on its own, come from? ____________________________ 3. How is this connected to the binomial expansion above? ______________________ 4. Pascal’s Triangle is closely tied to Combinations (which is why we look at it in this unit). The following chart should help us to find that tie… Row # # of terms Exponent on Binomial Coefficients of Binomial Expanded Using Combination Notation x y 0 = x y = 1 x y 2 x y 3 x y 4 x y n = = = = 31 Note n In the expansion of any binomial a b The coefficients of the terms are n C0 , n C1 , n C2 , …, n Cn The first term contains a nb0 and each subsequent term has powers of a that decrease by one and powers of b that increase by 1 until the last term is a 0bn The sum of the exponents on a and b is always n There are n 1 terms Example Write the first three terms in the expansion of 15 a. a b b. x 3y 12 Example Combine the above in order to expand: a) a b5 b) 3a 24 (5+1) or 6th row of Pascal’s triangle 32 c) x 26 d) 3 2 x 3 Note The Binomial Theorem (and, yes, it is on your formula sheet): t k 1 n Ck x nk y k where t k 1 is the term you are looking for (i.e. third, fourth) *the value for “k” is always one less than the term number (i.e. if you are looking for the third term,”k” is two) “n” is the exponent on the initial binomial “x” is the first term of the initial binomial “y” is the second term of the initial binomial Example Find the 8th term in the expansion of x y 14 33 Example Find the middle term in the expansion of 2 x 5 6 Example 8 The term with the y 5 in 5 2 y is… Example 11 1 Find the last term in the expansion of 3x 2y 34 Example 14 Find the term containing y 7 in the expansion of x 3 y . Example 14 1 Find the term containing x in the expansion of 2x3 . x 6 Example One term in the expansion of x a 8 is 448x6 . Determine the value of a, a 0 . 35 Example 6 1 Find the constant term in the expansion of x 2 . x Example 1 The constant term of 2 x 2 x 15 36 Example 12 2 Find the term independent of x in the expansion of 3x 5 . x Example 13 If the eleventh term of ax y is 2288x 3 y 10 , what is “a”? Textbook Page 542 #1ac,4ab,5ab,6ac,7abc,11ab,12a,17ad,18,19a,20 37 Permutation or Combination – Which is it? Always decide whether the question is a permutation or a combination first and then answer it by using the methods indicated with arrows on the previous page. 1. An airline pilot reported that in seven consecutive days she spent, in an unspecified order, one day at Winnipeg, one day in Regina, two days in Edmonton, and three days in Yellowknife. a) How many different itineraries are possible? b) How many itineraries are possible if she spent her first and last days in Yellowknife? 2. An executive consisting of a president, vice-president, treasurer and secretary must be formed from a group of 20 people. Calculate the number of executives possible. 3. A sub-committee of four must be formed from a group of 20 people. Calculate the number of sub-committees possible. 4. From a group of 5 student representatives, Albert, Bob, Carson, Deb and Evan, three will be chosen to work on a dance committee. a) How many committees are possible? b) List all the possibilities. 5. How many committees of three, with a chairperson, can be chosen from a group of 10 student representatives? Answers: 1. a) 420 2. 116 280 3. 4845 4. a) 10 5. 360 b) 60 b) ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE 38 Some More Practice Questions 1. In Class…Seven basketball players are sitting on the team bench. How many arrangements are there if the two forwards always sit together? (1440) 2. How many arrangements are there of the letters of the word JACOB if the vowels must be kept together? (48) 3. How many arrangements are there of the word CALGARY if the vowels must be kept together? (720) 4. In Class…What are the number of balanced meals that you can have if you have a choice of seven meats, four carbohydrates and five vegetables. You must choose one of each. (140) 5. There are two different ways that phone numbers in Okotoks may begin, either with 938 or 995. Other than that, any four digits may follow. How many phone numbers will have to be handed out in Okotoks before a new exchange is needed? (20000) 6. How many five card hands may be dealt from a standard deck that have two face cards? (652080) 7. How many five card hands may be dealt from a standard deck that have at least three black cards? (1299480) 8. How may three person “Special Math Committees” can be formed from a class of twenty students if the person with the highest average must be on the committee? (171) 9. In Class…There are seven teams in a league and each team must play each other team twice, once at home and once away. How many season games are there? (42 – two ways) 10. What is the term that contains the x 3 in the binomial expansion of 2 x 3 ? - 4320x 3 6 11. In Class…In the expansion of x a , one term is 352947x 2 , what is “a”? 7 12. In how many ways can the president, vice president and secretary be chosen from a class of twenty students? (6840) 13. How many arrangements are there of the word ARRANGEMENT? (2494800) 14. In Class…How many ways can you arrange the word ARRANGEMENT if your result must start with an “A”? (453600) 15. In how many ways can you arrange the letters of the word WELCOME if the result must start with a “C”? (360) 39 Permutation and Combination Review 1. If all the letters of the word DIPLOMA are used, then the number of different 7-letter arrangements that can be made beginning with three vowels is _________. 2. A car manager watns to line up 10 cars of identical model except for colour. There are 3 red cars, 2 blue cars and 5 green cars. Determine the number of possible arrangements of the ten cars if they are lined up in a row along one side of a parking lot and a blue car is parked on each end of the row. 3. A 6-player volleyball team stands in a straight line for a picture. If two particular players, Sam and Ellen, must be together, then how many different arrangements can be made for the picture? 4. A teacher tells his students that on a multiple-choice test with 12 questions, two answers are A, three are B, three are C and four are D. How many different answer keys are possible? 5. At a particular hotel, the following items are available for the continental breakfast: Beverage: coffee, tea, juice Pastry: muffin, toast, doughnut Fruit: apple, orange, grapefruit, banana If the continental breakfast consists of 1 beverage, 1 pastry, and 2 different types of fruit, then the number of possible breakfasts that can be ordered is: 6. A school committee consists of 1 vice-principal, 2 teachers, and 3 students. The number of different committees that can be selected from 2 vice-principals, 5 teachers, and 9 students is 7. In a basketball league, there are 6 teams. In the regular season, every team must play every other team twice. Determine the number of games that must be scheduled. 8. The vertices of an octagon are marked on a circle. Determine the number of triangles that can be formed using any three of the vertices. 9. In a group of 9 people, there are four females and 5 males. Determine the number of 4member committees consisting of at least one female that can be formed. 40 10. In a particular town, all the streets run north-south or east-west. A student must travel 5 blocks east and 3 blocks north to arrive at school. The number of different routes, 8 blocks in length, that the student can take to go to school is 11. Determine the coefficient of the term containing Ax 2 y 5 in the expansion of 2 x y . 7 12. Expand 2a 3 . 4 1 13. Determine the constant term of 2 x 2 x 6 14. A term of the binomial expansion ax y , where a 0 is 112 x 2 y 6 . The value of a, correct to the nearest whole number, is 8 ANSWERS: 1.144 2. 56 6. n 5 3. 240 7. 1680 11. 432 12. 56 5. 3 C1 3 C1 4 C2 4. 277200 8. 30 games 9. 56 13. A 84 14. 16a 4 96a 3 216a 2 216a 81 15. 60 16. a 2 17. 0.043 19. a) 34 b) 49 c) 240 18. a) 1 4 10. 121 b) 1 2 c) 7 8 41
