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Date:___________________________ Block:_______ Name:________________________________ Permutations and Combinations Introduction: In this unit we are going to examine how we can select or arrange a certain group of objects. I. Counting Problems A1. Consider the letters a, b, c, d. From this group we want to make as many different two letter “words” as possible. These “words” do not have to make sense and we may not use a letter twice in any one word. Consider all the “words” starting with ‘a’ - ab, ac, and ad. (Remember aa is not permitted since repeated letters are not permitted) Consider all the “words” starting with ‘b’ - ba, bc, and bd. (ba is not the same as ab just like no is not the same as on) Consider all the “words” starting with ‘c’ - ____, ____, and ____. Consider all the “words” starting with ‘d’ - ____, ____, and ____. How many words start with ‘a’? ____ with ‘b’? ____ with ‘c’? ____ with ‘d’? ____ How many different words are there altogether? ____ A2. Consider 3 cars racing: a Ford, a Dodge, and a Buick. How many different ways may they finish 1st and 2nd? 1st place 2nd place Ford Ford How many different ways are there? ____ Dodge Dodge Buick Buick A3. If you order a sundae in a certain ice cream parlor, you may pick one scoop of ice cream from vanilla, chocolate, strawberry or peppermint and top the sundae with one kind of syrup from butterscotch, marshmallow or cherry. How many different types of sundaes could you order? Are there a correct number of blanks? Is your answer 12? 1 Date:___________________________ Block:_______ Name:________________________________ A4. The positions of president and vice-president of a corporation are vacant and only 2 men, Mr. Expansion and Mr. Tightwad are eligible for either position. List the ways in which the positions can be filled. How many ways did you get? _____ A5. How many different signals could be made from a red, a green, and a blue flag if all three flags must be used? List them. II. Tree Diagrams So far the answers have been easily found by listing all the possible arrangements but it should be obvious that this could become very long and boring in many cases. For example: How many 7 digit phone numbers can we make from the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9? If you are not convinced that this is long, try it! There must be an easier way to solve this type of problem. B1. Lets go back to the problem A1 with the letters a, b, c, d and build what is called a TREE. 1st letter 2nd letter How many choices did we have for our 1st letter? b ______(a, b, c, d) a c No matter which letter we choose for our 1st letter, d how many choices did we have for our second letter? _____ a Can you see a way to relate the number of choices b c for our 1st letter and the number of choices for the d second letter and the total number of “words”, 12? ____________ ____________ =______________ a # of choices for # of choices for # of possible c b a 1st letter a 2nd letter different “words” d What operation did you use between the 1st and 2nd a blank? ______________ d b c B2. Let’s consider example A2. How many different choices of winners could we have? How many possible 2nd place choices are there? Since we saw earlier that there are 6 possible ways for the cars to finish, does the relationship that you saw on the bottom of the previous page still apply? _______________ # of choices for a winner ________________ # of choices for 2nd place = ______________ # of ways the cars can finish What operation did you use between the 1st and 2nd blank? _____________ 2 Date:___________________________ Block:_______ Name:________________________________ B3. In problem A3, you had ______ choices of ice cream. For each choice of ice cream, how many choices of syrup were there? ______ Thus, how many choices of sundaes are there? ______ _______________ x _____________ = ____________ B4. For example A5, how many choices for the first flag are there? _________ For each 1st flag, how many choices for the 2nd flag are there? ________ For each 2nd flag, how many choices for the 3rd flag are there? ________ _________ x _________ x _________ = ___________ The number of possible signals is ______________ III.The Fundamental Counting Theorem The pattern we have seen on the previous page is a short cut ( as opposed to actually listing) for counting called the fundamental counting theorem. It says: To find the number of ways of making several choices in succession, multiply the number of choices that can be made in each successive decision. IV. Problems using the Fundamental Counting Theorem 1. In how many ways can all the letters of the word “sail” be written? _________ x _________ x __________ x __________ = __________ 2. In how many ways can the captain and co-captain of a team be chosen from 11 players? 3. In how many ways can the 4 infield positions of a baseball team be filled if there are 6 candidates for the position? 4. In how many ways can the letters of the word “frisby” be used to write, without repeating letters? a) 2 letter words ________ b) 3 letter words___________ c) all letter words ___________ 5. Linus is taking a true or false test, which has only 5 questions. He seems to be guessing at every answer. How many different patterns of true or false are there for the five questions? __________x__________x__________x__________x__________ = __________ 7. You have 7 sweaters, 5 pairs of slacks and 10 shirts. If you plan to wear a shirt, a sweater and a pair of slacks everyday, how many different outfits can you make? _______ 8. You know that there are _____ digits in our number system, 0, 1, 2, …8, and 9. How many different 3-digit numbers can we make if the first digit can’t be a zero and no digit can be repeated? 3 Date:___________________________ Block:_______ Name:________________________________ Answers to section IV (p 3) 1. 4x3x2x1 = 24 2. 11x10 = 110 3. 6x5x4x3 = 360 4.a) 6x5 = 304.b) 6x5x4 = 120 4.c) 6x5x4x3x2x1 = 720 5. 2x2x2x2x2 = 32 7. 7x5x10 = 350 8. 9x9x8 = 648 V. Permutations The definition of a permutation is: The arrangement of certain objects in a particular order is called a permutation. Some of the problems on the previous pages have been permutations and some have not. For example: In sections IV #4 is a permutation. The “word” fy is different from the “word” yf because the order, in which the elements f and y appear, differs. Section IV, Example #6 is not a permutation because having John and Sally or Sally and John, as a couple would be the same. The order chosen doesn’t make a different couple. Notation: n Pr Before we explain the strange symbol above, solve the following permutation. How many ways can the 20 people in a club be chosen to be President, Vice-president, Secretary, and Treasurer? ___ x ___ x ___ x ___. Mathematicians would call this a permutation of 20 elements taken 4 at a time. They use the symbol 20 P4 . The order chosen is important. John for president and Sally for vice-president, is different from Sally for president and John for vice-president. In general , n Pr means that you have “n” numbers and you plan to pick “r” of them. 1. a) Find 10 P3 . 10 P3 = 10 x 9 x 8 =720 Do you see a pattern? b) Show that 6 P4 = 360. ___ x ___ x ___ x ___ = ____ c) 7 P2 = _______ d) 50 P3 = ________ n) i. How many ways can the letters of the word PENCIL be formed into 2 letter “words” if there are no repetitions? ____P____ =_____ f) 4 P4 = _______ h) 6 P2 = _______ j) 100 P2 = _______ l) n1 P3 = _______ m) Solve for “n” if n1 P2 30 . _____ x _____ = 30 e) 8 P4 = ________ g) 99 P1 = _______ i) 6 P4 = _______ k) n P2 = _______ 4 ii. How many 3 letter words can be formed from PENCIL? ____P____ = ________ iii. How many 6 letter words can be formed from PENCIL? ____P____ = ________ o) What is wrong with 6 P7 ? ____________ p) Five people are to stand in line. In how many ways can they be arranged? Date:___________________________ Block:_______ Name:________________________________ The symbol n! ( Factorial Form) Problems like p) above which was a permutation of 5 things taken 5 at a time occur quite often. Problems that use numbers like 5 P5 , 17 P17 , 4 P4 are given a special symbol and name. P4 = 4 x 3 x 2 x 1 is written 4! It is read “4 factorial”. Note that it is the product of all the integers 1 through 4. 7 P7 = 7 x 6 x 5 x 4 x 3 x2 x1 = 7! 2A) What would be the factorial form for 6 x 5 x D). How many 8 digit #’s can you make using the 4 x 3 x 2 x 1? ______ digits 9, 8, 7, 6, 5, 4, 3, 2? Leave your answer in factorial form. _____ B) 12 P12 = ____! 4 C) In how many ways can the letters A, B, C, D, E, F be arranged 6 at a time? Write your answer using both symbols you have learned. ___P___ and _____! Answers to Section V 1. c) 7 x 6 = 42 d) 50 x 49 x 48 = 117600 e) 8 x 7 x 6 x 5 = 1680 f) 4 x 3 x 2 x 1 = 24 g) 99 h) 6 x 5 = 30 i) 6 x 5 x 4 x 3 =360 j) 100 x 99 = 9900 k) n x (n-1) l) (n+1) x (n) x (n-1) = n 3 n m) (n+1)(n) = 30 n 2 n 30 0 (n 6)( n 5) 0 Only the positive answer is acceptable. n 6, n 5 n) i. 6 P2 ii. 6 P3 iii. 6 P6 o) You cannot take 6 digits 7 at a time. p) 5 P5 2.a)6! b)12! c) 6 P6 = 6! d) 8 P8 = 8! VI. Permutation Problems Use permutation notation to express the answers, n Pr or n! and also show the factors. Examples: 1. How many ways can 7, 5, 2, 3, 1, 8 be made into a 4 digit number? Answer: 6 P4 = 6 x 5 x 4 x 3 2. How many ways can 7 people line up to buy tickets at the movies? Answer: 7 P7 = 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 5 Date:___________________________ Block:_______ Name:________________________________ Problems 1. How many ways can a daily schedule of 5 2. How many ways can 4 people be assigned seats classes be selected if 20 different classes are if there are 10 seats available? offered? 3. Twelve students have qualified for 3 awards. 4. In a horse race you are going to pick one horse In how many ways can the prizes be given out if a to win, one horse to place, (2nd place) and one student may only receive one award? horse to show (place 3rd). If there are 15 horses in the race, how many ways can you make your selection? 5. In how many ways can the letters A, B, C, D, E, F, G be arranged to make: a) 5 letter “words”? b) 7 letter “words”? c) 1 letter “words”? 6 Using the letters from the word MONTREAL, a) How many 6 letter “words” can be made? b) How many 7 letter “words” can be made? c) How many 8 letter “words” can be made? d) If we said how many “words” can be made with at least 6 letters, we would all the possibility of having 6 letter “words” or ___ letter “words” or ___ letter “words”. Thus the answer is ____ + ____ + ____ = _____ 7. The Greek alphabet consists of 24 letters. How many Greek letter fraternity names can be formed; a) containing 3 different letters? _____ b) containing 2 different letters? _____ c) containing only 1 letter? _____ d) containing no more than 3 letters? _____ The next set of problems will have some special conditions involved. Take care of the special condition first, then the rest of the problem. If you can simplify your answer by using n Pr notation, then do so. Example: How many even 4 digit numbers can be made from the numbers 1, 3, 4, 5, 6, 7, 9 if no repetition is allowed? Answer: We want a four digit number where the last number must be divisible evenly by 2. We have only 2 choices for this last spot, 4 or 6. Let the spaces below signify how many choices we have to fill each location in our 4 digit number. # of __________ __________ __________ __________ choices for 1st number choices for 2nd number choices for 3rd number choices for last number The blanks would be filled in as follows: 6 x 5 x 4 x 2 = 6 P3 x 2 (The last space was filled first with a 2. This reduced the number of choices for the first space from 7 to 6 etc.) 6 Date:___________________________ Block:_______ Name:________________________________ 8. From the letters a, b, c, d, e, f, 3 letter “words” are to be made. How many “words” can be formed if it must not begin with “b”? Remember to take care of the restriction first. 9. How many of the arrangements of all the letters of the word EQUATION begin with a vowel? 10. How many license plates can be formed if each consists of 2 different letters from the alphabet and 5 different digits? _____x_____ x _____x_____x_____x_____x_____ or 26 P____ 10 P____ Letters Digits 11. How many license plates can be formed if each contains 2 letters and 5 digits and repetitions of both are permitted? From this point onward assume that repetitions are not allowed unless otherwise stated. 12. How many 7 digit phone numbers can be made using any of the digits 0 through 9, if the phone company does 14. How many odd 5 digit numbers can be formed using the digits of the number 3,710,684? Remember that the first digit will not be a zero. a) Not wish the number to begin with zero, and no numbers can be repeated? b) Not wish the number to begin with 0 or 9, and no repeats allowed? c) Not wish the numbers to begin with 0 or 9 and numbers may be repeated? 15. How many 3 digit numbers can be formed from the digits 2, 3, 4, 5, 6, 7, 8, 9 with no repetitions if: a) the first two may not be a 2 or a 3? b) the number may not be divisible by 5? 16. The dial of a combination lock has 26 letters of the alphabet on it. If a combination is formed by dialing 4 different letters in a particular order, a) How many combinations are possible? 13. How many 6 letter words can be made from the letters of the word DOCILE so that the first and the second are vowels? b) If the owner forgets his combination, what is the maximum number of wrong 4 letter combinations he could dial before getting the correct one? Answers to SECTION VI 1. 20 5.a) P5 20 19 18 17 16 7 P5 7 6 5 4 3 2. 10 P4 10 9 8 7 b) 7 P5 7! 3. 12 P3 12 11 10 4. 15 P3 15 14 13 c) 7 P1 7 6. a) 8 P6 8 7 6 5 4 3 P2 24 23 c) 24 P1 24 d) 24 P1 + 24 P2 + 24 P3 8. 5 5 4 55 P2 9. 57 P7 5 7! 10. 26 P2 10 P5 11. 26 26 10 10 10 10 10 12. a) 9 9 8 7 6 5 4 99 P6 b) 8 9 8 7 6 5 4 89 P6 b) 8 P7 8 7 6 5 4 3 2 c) 8 10 6 13. 3 24 P4 16.a) 26 P4 b) 26 P4 1 c) 8 P8 8! d) 8 P6 8 P7 8 P8 7.a) 14. 5 5 4 3 3 24 P3 24 23 22 b) 15.a) 6 5 6 66 P2 7 24 b) 7 6 7 Date:___________________________ Block:_______ Name:________________________________ VII Another way to write n Pr Before we move on to another specific type of counting problem, called COMBINATIONS, we need to examine n Pr and n! notation more carefully. There is a relationship that needs to be understood and we will use this relationship when we discuss combinations. Recall that 5! = ____x____x____x____x____. With this knowledge, simplify the following. Do not multiply your answers out. 12! = = 12 1110 9 8 7 6 5! 5 4 3 2 1 10! 24! 15! 2. = 3. = 4. = 7! 20! 9! 1. 5. 6! = 2! 8! without writing down all the factors 5! and reducing? _______ Try to write down in words the pattern that you see. Do you see a pattern? ________ Can you give the answer to Do the following problems using your pattern. 6. 100! = 95! 7. 6! = 4! 8. 58! = 57! 9. 37! = 30 10. Can you go back to page 10 and write your answers in n Pr form? For example the answer to question 1 was 12 1110 9 8 7 6 = 12 P7 2. _____ 3. _____ 4. _____ 5. _____ 6. _____7. _____ 8. _____ 9. _____ 11. a) 5! 3! b) (n 2)! = n! 12. Solve for all n which satisfy: (n 1)! (n 2)! 20 a) b) 72 n! (n 1)! n! 13. We have seen that given we can write the problem in permutation notation. Try to generalize. m! n! = ___ P___ m! 14. Let’s go the other way. Given the following permutation notations, write them as the division of 2 factorials. a) 16 P2 h) 5 P5 b) 10 P3 i) n P5 c) 100 j) 10 P70 Pn 15. In question 14 h) we found 5 P5 d) 25 P17 e) k) n Pr 50 P24 f) 17 P3 l) n r P1 5! and 5 P5 also equals 5 4 3 2 1 5! 0! So 0! Must be defined as equal to one 8 g) 13 P1 Date:___________________________ Block:_______ Name:________________________________ ANSWERS TO SECTION VII 1. 12 1110 9 8 7 6 2. 10 9 8 3. 24 23 22 21 4. 15 14 13 12 1110 5. 7. 6 5 8. 58 9. 37 36 35 34 33 32 31 6 5 4 3 6. 100 99 98 97 96 10. 2) 10 P3 3) 24 P4 4) 15 P6 5) 6 P4 6) 100 P5 7) 6 P2 8) 58 P1 9) 37 P7 11. a) 20 b) (n 2) (n 1) 12.a) n=8 (Discard n=-9) b) n=3 (Discard n=-6) 13. n P( n m ) n! 10! 16! 10! 100! 25! 50! 17! 13! 5! b) c) d) e) f) g) h) i) j) 14! 7! 8! 26! 14! 12! 30! 0! (n 5)! (10 n)! (n r )! n! k) l) (n r 1)! (n r )! 14.a) VIII. Combinations A) Definition of a combination Permutation problems involve a selection or arrangement where the order in which the object is chosen is important. There are other types of problems where the order in which the object is chosen is not important. This type of problem is called a combination. B) Notation n C r We have used the notation n Pr to say we want the number of permutations (order is important) of 5 objects selected 2 at a time. 5 C 2 says that we want the number of combinations (order is not important) of 5 objects selected 2 at a time. In general, n C r says we want the number of combinations of n objects selected r at a time. C) Problems involving combinations 1. We want to know how many different committees with 2 people on them can be formed from 4 people, Mary, Jeff, Sarah and Tom. The permutations of these 4 people chosen 2 at a time are listed below: Mary, Jeff Mary, Sarah Mary, Tom Jeff, Mary Sarah, Mary Tom, Mary Tom, Mary Tom, Jeff Tom, Sarah Mary, Tom Jeff, Tom Sarah, Tom We have 12 different permutations of 2. With permutations the order selected is important, thus Mary, Jeff is different from Jeff, Mary. We, on the other hand, want committees of 2. Does order matter when selecting committees? If order does not matter, we have duplications in our list above. Mary, Jeff would be the same committee as Jeff, Mary. Cross out the duplications in your list above ant then count how many committees of 2 are left. You should now have only 6 committees. Thus we see that 4 C2 = 6, while 4 P2 = 12. 9 Date:___________________________ Block:_______ Name:________________________________ Explain why there are twice as many permutations as there are combinations. Discuss it with friends and possibly your teacher 2. You are making a present to send to a friend. You want to send 4 objects which are to be selected from 1 orange, 1 apple, 1 plum, 1 pie, and 1 cake. The 5 different packages you could send are listed below: orange, apple, plum, pie orange, apple, plum, cake orange, apple, pie, cake orange, pie, plum, cake apple, pie, plum, cake Since order doesn’t matter when filling the package, the package orange, apple, plum, cake is the same as the package apple, plum, cake, orange. We didn’t list the duplications. This type of problem is a combination because the order of selection wasn’t important. We have found that 5 C 4 5 . Why is the answer to this problem a “C” notation instead of a “P” notation? What does 5 P4 equal? Does it seem logical that 5 C 4 should be less than 5 P4 ? D) Evaluating n C r We evaluate the number of combinations by the rule: a) 10 C8 b) 30 C3 n Cr = n! r!(n r )! 10! 10 9 8 7 6 5 4 3 2 1 10 9 90 45 8!2! 8 7 6 5 4 3 2 1 2 1 2 1 2 30! 30 29 28 27! 30 29 28 5 29 28 4060 3!27! 3 2 1 27! 6 c) 10 C5 d) 100 C 98 = e) 100 C2 = f) 30 C 2 g) 30 C 28 h) 25 C 24 i) 12 C9 j) 7 C 7 = n! is actually the number of permutations divided by the number of r!(n r )! permutations within each combination. Making this division eliminates the duplications in each permutation Note: The rule for n C r ANSWERS TO SECTION VIII c) 252 d) 4950 e) 4950 IX. f) 435 g) 435 h) 25 i) 220 j) 1 Combination Problems The important thing to remember when trying to distinguish between combination and permutation problems is: does the order in which the items are chosen matter? If the answer is no, you are dealing with a combination problem. 1. For example: How many ways can Sally, Jane, and Tom form a committee of two members? Think to yourself “Is the order chosen important?” The answer is no since Sally and Tom are the same committee regardless of who was chosen first. We will calculate the number of different committees by evaluating 3 C 2 which numerically equals ____________. If we asked how many ways could a 10 Date:___________________________ Block:_______ Name:________________________________ chairman and an assistant chairman be chosen, then Sally as chairman and Tom as assistant chairman is completely different from Tom as chairman and Sally as assistant chairman. We would then have a permutation and 3 P2 would be used. Numerically this would be____________. Note that n Pr is always equal to or greater than n C r 2. In how many ways can a committee of 3 be chosen from Jeff, Stan, Barb and Mary? ____! ____ ____ C ____ = ____! ____! 3. How many flight crews of 15 aircraft can be selected if there are 18 planes to choose from? 7. In how many ways can a team of debaters be chosen to represent CHS if there are 6 people eligible and: a) 3 are chosen? _______________ b) 4 are chosen? _______________ c) 5 are chosen? _______________ d) 6 are chosen? _______________ e) at least 3 are chosen? _________ 4. In how many ways can a clean up crew of 6 be chosen if 14 people volunteer? 8. In how many ways can a “Reach for the Top” team be chosen from 4 grade 12’s and 6 grade 5. This school offers 12 different courses to grade 11’s if the team has: 12 students. How many different lists of 5 classes a) only four members? ___________ b) 4 members exactly one of which must be in could you give your guidance counselor? grade 11? ___________ c) 4 members with no more than 1 grade 12 6. a)In how many ways can a coach reject sufficient players to cut his team down from 15 to member? ___________ d) ast 2 grade 12 members out of the 4? ______ 11 players? b) In how many ways can a coach select 11 of 15 players to go to a tournament? Compare the numerical answers to parts a) and b) in question 6. Try to explain why they are the same. 9. In how many ways can an athletic board of 6 people be chosen from 6 teachers and 10 students if the board must have: a) exactly 2 teachers? ___________ b) exactly 1 teacher? ___________ c) at least 3 teachers? __________ d) no teachers? __________ ANSWERS TO SECTION IX 2. 4 C3 4 3. 18 C15 3 17 16 4. 14 C6 7 13 11 3 5. 12 C5 12 11 3 2 6.a) 15 C 4 15 7 13 b) 15 C11 15 7 13 7.a) 6 C3 b) 6 C 4 c) 6 C5 d) 6 C3 6 C4 6 C5 6 C6 8.a) 10 C 4 b) 6 C1 4 C3 c) d) 4 C2 6 C2 4 C3 6 C1 4 C4 6 C0 9.a) 6 C 2 10 C 4 b) 6 C1 10 C5 c) 6 C 4 4 C1 6 C3 d) 10 C 6 6 C3 10 C3 6 C 4 10 C 2 6 C5 10 C1 6 C6 11 Date:___________________________ Block:_______ Name:________________________________ X. Random Counting Problems Some of these problems are permutations, some are combinations, some just require plain common sense. Remember, to answer with a permutation, the order of chosen items must be important! 1. Peter has 5 books he wants to read, but the library will only let him check out 3. How many sets of books could he take home with him? (Is order important when checking out books?) 2. Peter doesn’t know in what order to read the books. How many different ways could he read the 3 books? 3. How many ways can a pilot and co-pilot be chosen from 8 candidates? 4. How many ways can a crew of 5 stewards be chosen from a pool of 6? 5. In how many ways can a flight of 15 bombers be chosen if there are 20 aircraft available and the aircraft of the squadron leader must always be chosen? 6. The post office plans to issue new 10, 20, 30, 40, and 50 cent stamps. Each denomination must be of a different colour. If there are 11 colours available, in how many ways can the colours be chosen? 7. In how many ways can the letters of the word PRODUCT be written to form 7 letter “words” beginning with: a) a vowel? b) a consonant? 8. How many ways can a jury of 7 people be chosen from 4 women and 10 men if it must have: a) exactly 3 men? b) at least 3 women? c) the number of men and women doesn’t matter? ANSWERS TO SECTION X 1. 5 C3 10 2. 3 P3 6 3. 8 P2 56 4. 6 C5 6 5. 19 C14 11628 6. 7.a) 26 P6 1440 b) 56 P6 3600 8.a) 10 C3 4 C 4 120 b) 4 C3 10 C4 4 C4 10 C3 960 c) 14 C 7 UP NEXT PROBABILITY!... 12 11 P5 55440
