> Chapter 10 Permutations and combinations THIS SECTION WILL SHOW YOU HOW To: find the number of arrangements of » distinct items. find the number of permutations of r items from n distinct items find the number of combinations of r items from n distinct items solve problems using permutations and combinations.10 Permutations and combinations 10.1 Factorial notation 5x 4x3 x2 | iscalled ‘5 factorial’ and is written as 5! permutations combinations Itis useful to remember that n! nx(a-l For example, 5! = 5 * 4! Find the value of 8! BL 8X7XOX IK AX IKIRY FRIR IRIEL =8x7x6 = 336 fp HL x10x9K BRT XW EXAAIND RY BIBL RK TK MX HK AK IX IX YXIX2x1 = 290 “so = 165 Exercise 10.1 1 Without using a ealculator, find the value of each of the following, Use the x! key on your calculator to check your answers » a n a 7 ba ee a a a ® a am pa 2a ‘S307 ay? 2 Rewriteeach ofthe following using factorial notation a 2x1 b 6xSx4x3x2K1 € Sxdx3 Ox 9x8 [2x UK WO Ke @ 17% 16% 1514 © 3x2x1 t aKa Rex 3 CHALLENGE QUESTION Rewrite cach of the following using factorial notation, a n(n~1)(a-2)(n-3) b n(n ~ 1) 2)(n— 3) (= 4)(n- 5) no ~1)(n=2) SHOR IRIRE 231)> CAMBRIDGE IGCSE™ AND 0 LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK 10.2 Arrangements These books are arranged in the order BROG. (Blue, Red, Orange, Green). The books could be arranged in the order OGBR. Find the number of different ways that the 4 books can be arranged in a line. ‘You will need to be systematic. How many ways are there of arranging five different books in a line? To find the number of ways of arranging the letters A, B and C in a line you can use two methods. Method 1 List all the possible arrangements, ‘These are: ABC ACB BAC BCA CAB and CBA. ‘There are 6 different arrangements, Method 2 Consider fing 3 spaces. O ‘The first space can be filled in 3 ways with either A or B or C. For each of these 3 ways of filling the first space, there are 2 ways of filling the second space. ‘There are 3 x 2 ways of filling the frst and second spaces. For each of the ways of filling the first and second spaces there is just 1 way of filling the third space. ‘There are 3 x 2% 1 ways of filling che Ctee spaces. ‘The number of arrangements =3 x 2 1=6 3 x 2x | iscalled °} factorial” and can be written as 3! In the class discussion, you should have found that there were 24 different ways of arranging the 4 books. 4)=4x3x2x1=24 ‘The number of ways of arranging m distinct items in10 Permutations and combinations Poneevmure GHeOUEeR8 a Find the number of different arrangements of these nine cards, if there are no restrictions. b Find the number of arrangements that begin with GRAD. ¢ Find the number of arrangements that begin with G and end with S. Answers a Thereare 9 different cards. number of arrangement 362880 b The first four letters are GRAD, so there are now only 5 letters left to be arranged. number of arrangements = 5! = 120 ‘¢ The first and last letters are fixed, so there are now 7 letters to arrange between the G and the S. number of arrangements HHRHOOCOe a Find the number of different arrangements of these seven objects if there are no restrictions. \d the number of arrangements where the squares and circles alternate. ¢ Find the number of arrangements where all the squares are together. d__ Find the number of arrangements where the squares are together and the circles are together. Answers a There are 7 different objects. ‘number of arrangements = 7! = 5040 b If the squares and cireles alternate, a possible arrangement is: OnenOn ifferent ways of arranging the four squares. ‘There are 4! There are 3! different ways of arranging the three circles. So, the total number of possible arrangements = 4! x 3! = 24 x 6= 144> CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK If the squares are all together, « possible arrangement is: ‘The number of ways of arranging the | block of four squares and the 3 circles = 41 There are 4! ways of arranging the four squares within the block of squares. So, the total number of possible arrangements = 4! x 4! = 24 x 24 = 576 If the squares are together and the circles are together, a posible arrangement is: ‘There are 4! x 3! ways of having the squares atthe start and the circles at the end. Another possible arrangement is: ‘There are 3! x4! ways of having the circles atthe start and the squares atthe end. total number of arrangements = 4! x 3!+ 3! x 4! Exercise 10.2 1 Find the number of different arrangements of @ 4 people sitting in a row on a bench b 7different books on a shelf, 2 Find the number of different arrangements of letters in each of the following words a TIGER b OLYMPICS ¢ PAINTBRUSH 3 @ Find the number of different four-digit numbers that can be formed using the digits 3, 5, 7 and 8 without repetition. b How many of these four-digit numbers are i even 4 A shelf holds 7 different books. Four of the books are cookery books and three of the books are history books. @ Find the number of ways the books can be arranged if there are no restrictions. b Find the number of ways the books can be arranged if the 4 cookery books are kept together. 5 Five-digit numbers are to be formed using the digits 2, 3, 4, 5 and 6. Bach digit may be used only onee in any number. @ Find how many different five-digit numbers can be formed. greater than 8000?10 Permutations and combinations How many of these five-digit numbers are b even © greater than 40000 devon and greater than 40000? 6 Three girls and two boys are to be seated in @ row. Find the number of different ways that this can be done if a the girls and boys sit alternately b agirl sits at each end of the row © the girls sit together and the boys sit together. 7 @ Find the number of different arrangements of the letters in the word ORANGE. Find the number of these arrangements that b begin with the letter © © have the letter O at one end and the letter E at the other end. 8 a Find the number of different six-digit numbers which can be made using the digits 0, 1,2, 3, 4 and 5 without repetition. Assume that a number cannot begin with 0. b How many of the six-digit numbers in part a are even? 9 Six girls and two boys are to be seated in a row. Find the number of ways that this can be done if the two boys must have exactly four girls seated between them, 10.3 Permutations Inthe last section, you learned that if you had three leters A, B and C and 3 spaces to fill, then the number of ways of filing the spaces was 3 ¥ 2* 1 = 3! ‘Now consider having & letters A, B, C, D, E, F, G, H and 3 spaces to fill. ‘The first space can be filled in 8 ways For each of these 8 ways of filling the first space, there are 7 ways of filling the second space, There are 8 x 7 ways of filling the first and second spaces. For cach of the ways of filling the first and second spaces there are 6 ways of filling the third space. ‘There are 8 x 7 6 ways of filling the three spaces, ‘The number of different ways of arranging three fetiers chosen from eight letters 8x 7x 6=336. “The different arrangements of the letters are called permitations. ‘The notation ‘P, is used to represent the number of permutations of 3 items chosen from 8 items.> CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK 8x Tx 6x5x4x3x2% Sx4x3K 2x1 8! Note that 8 * 7 6 can also be written as 8! So's = Gay ‘The general rule for finding the number of permutations of items from ngistnetitemsis"?, = 7" Note: ‘+ Inpermutations, order matters. © By definition, 0! = 1 To explain why 0! = 1, consider finding the number of permutations of 5 letters taken from 5 letters. oe LS choices choices choices choices choke ‘The number of ways of filling the 5 spaces with the 5 letters = 5 x 4 x 3x2 1 = 120, josie, 120 But = G5 O01 sq 120 So 130-= 120, Hence 0! must be equal to | A security code consists of 3 leters selected from A, B, C, D, B, F followed by 2 digits selected from 5, 6 7,8 Find the number of possible security codes if no letter or number can be repeated. Answers Method 1 ‘There are 6 letters and 5 digits to select from. ‘Number of arrangements of 3 letters from 6 letters = °P ‘Number of arrangements of 2 digits from 5 digits = *P, So, the number of possible security codes = *P, x °; ‘Method 2 There are 6 letters and 5 digits to select from, ee choices choices choices choices choices 236)10 Permutations and combinations ‘The first three spaces must be filled with three of the six letters ‘There is a choice of 6 for the first space, 5 for the second space and 4 for the third space. The last two spaces must be filled with two of the 5 digits ‘There isa choice of 5 for the first space and 4 for the second space. So, the number of possible security codes = 6 x 5x 4% 5 x 4= 2400. Mela Find how many even numbers between 3000 and 4000 can be formed using the digits 1, 3, 5,6, 7 and 9 if no number can be repeated. Answers Method 1 ‘The first number must be a 3 and the last number must be a 6. ‘There are now two spaces to fill usi 1g two of the remaining four digits 1, 5, 7 and 9, Number of ways of filling the remaining two spaces = “P, = There are 12 different numbers that satisty the condi ‘Method 2 Consider the number of choices for filling each of the four spaces. ions. fp choice choices choices choive ‘Number of ways of filling the four spaces = 1 x 4 x3 1 ‘There are 12 different numbers that satisfy the conditions. ‘Method 3 In this example it is not impractical to list all the possible permutations, These are: 3156 3516 3176 3716 3196 3916 3576 3756 3196 3916 3796 3976 ‘There are 12 different numbers that satisty the conditions. 237)> CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK 2. (1-605) =0 n=6 orn nis a positive integer "Ps= Tala = = 2) 1X ~ Vin = Din = 3)n = Ain ~ 5) 5 n(n In ~ 2y(n— 3)(n— 4) = T2nG~ Iin= 2) divide both sides by nn ~ 10 ~ 2) = Poni ~ Win = 2) (2 3in=4) m-7n-60=0 Dyin 5 1nis.a positive integer Exercise 10.3 1 Without using a calculator, find the value of each of the following, Use the "P, key on your calculator to check your answers. a Py b Py co "Ry a) By 2 Find the number of different ways that 4 books chosen from 6 books be arranged ona shelf. 3 How many different five-digit numbers can be formed from the digits 1, 2,3, 4,5, 6,7, 8,9 if no digit can be repeated? 4 There are 8 competitors in a long jump competition, In how many different ways can the first, second and third prizes be awarded? 5 Find how many different four-digit numbers greater than 4000 that can be formed using the digits 1,2, 3,4, 5, 6 and 7 if no digit can be used more than once, 6 Find how many even numbers between 5000 and 6000 can be formed from the digits 2, 4,5, 7, 8, if no digit can be used more than once.410 Permutations and combinations 7 A four-digit number is formed using four of the eight digits 1, 2, 3,4,5,6, 7and 8 No digit can be used more than once, Find how many different four-digit numbers can be formed if there are no restrictions b the number is odd © thenumber is greater than 6000 d__ the number is odd and greater than 6000. 8 Numbers are formed using the digits 3,5, 6, 8 and 9. No digit can be used more than once. Find how many different a three-digit numbers can be formed b numbers using three or more digits can be formed. 9 Find how many different even four-digit numbers greater than 2000 can be formed using the digits 1, 2, 3, 4, 5, 6, 7, 8 if no digit may be used more than once. 10 a Solve"P,= 720 b Solve"Ps = Tnin~ 1) Ps © Solvesy” EUCainelssron! You have already investigated the number of ways of arranging 4 different books in a line. You are now going to consider the number of ways you can select 3 books from the 4 books where the order of selection does not matter. If the order does not matter, then the selection BRO. is the same as OBR. Find the number of different ways of selecting 3 books from the 4 books. A combination sa selection of some items where the order of selection does not matter. So, for combinations order does mor matter. Consider the set of 5 crayons. ‘To find the number of different ways of choosing 3 crayons from the set of 5 crayons you can use two methods.> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK ‘Method 1 List all the possible selections. ‘There are 10 different ways of choosing 3 crayons from 5. “Method 2 sx4 i = dL 2 SK4_ ‘The number of combinations of 3 from $= 3>="3,~= 10 10.4 Combinations Svar, ‘A team of 6 swimmers is to be selected from a group of 20 swimmers. Find the number of different ways in which the team can be selected. Answers Number of ways of selecting the team = "C, = GE = 38.76010 Permutations and combinations ‘The diagram shows 2 different tents A and B. A B ‘Tent A holds 3 people and tent B holds 4 people, Find the number of ways in which 7 people can be assigned to the two tents. Answers ‘Number of ways of choosing 3 people from 7 for tent A = 'C; = 35. So, the number of ways of assigning the 7 people to the two tents = 35. 3 coats and 2 dresses are to be selected from 9 coats and 7 dresses, Find the number of different selections that can be made, Answers. Number of ways of choosing 3 coats from 9 coats, Number of ways of choosing 2 dresses from 7 dresses = " Cy °Cy x "Cy = 84 x 21 = 1764 So, the number of possible selections A quiz team of 5 students is to be selected from 6 boys and 4 girls. Find the number of possible teams that can be selected in which there are more boys than girs, Answers If there are more boys than girls there could be: S boys and 0 girls number of ways = °C, x “C) =6 x 1= 4 boys and 1 giel mumber of ways = $C, x *C, = 15 x4 = 60 3 boys and 2 girls number of ways = °C x *C, = 20x 6 = 120 ‘The total number of possible teams = 6 + 60 + 120= 186> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK Sofia has to play 5 pieces of music for her music examination, She has 13 pieces of music to choose from. ‘There are 7 pieces written by Chopin, 4 written by Mozart and 2 written by Bach Find the number of ways the 5 pieces can be chosen if a there are no restrictions b there must be 2 pieces by Chopin, 2 pieces by Mozart and | piece by Bach there must be at least one piece by each composer. Answers a Number of ways of choosing 5 from 13 = '""C; = 1287 ‘Number of ways of choosing 2 from 7 pieces by Chopin = "C ‘Number of ways of choosing 2 from 4 pieces by Mozart = “C; Number of ways of choosing | from 2 pieces by Bach =°C, So, number of possible selections = °C, x *C, x 7C, = 21 x 6 x 2= 252 © If there is at least one piece by each composer there could be: 3Chopin Mozart 1 Bach number of ways = ’C, x “C, x °C, = 35 «4x 2=280 1 Chopin 3Mozart 1Bach number of ways = 'C, x “C, x °C, = 7x 4x 2=56 2Chopin 2Mozart 1 Bach number of ways = "C; x “C, x °C, = 21 x 6% 2= 252 2Chopin Mozart 2Bach umber of ways = 7C, x “C, x 7C, = 21 «4x 1 = 84 1Chopin 2Mozart 2Bach number of ways = 'C, x “C, x*C,=7*6«1=42 Total number of ways = 280 + 56 +252 + 84+42= 714 Given that 7 * "C3 = (n~ 18) x "'C,, find the value of n, Answer "C2 nl DOG TX a Pig = pipe ee 0-18) x". = 0-19) * GoTo ew! - 0-19) BH replace (n+ I)! with (n+ 1) x nt cE (n+ 1) xnt . _ =~ 18) *Frareq= sy ePCe with 6% 5110 Permutations and combinations SN vs5) @rixt Zi — 1) rege Mie both sides by (180+ 1) 7 6 42=(n- 18141) 42= 08 11n-18 170 60=0 (n-2)(0¥3)=0 20orn=-3 mis. positive integer 20 Hence 7 nl/5t(n= 5)! Exercise 10.4 1 Without using a calculator, find the value of each of the following, and then use the “C, key on your calculator to check your answers. « 4 8 5 1 at bt, ey af) 2 § + 0 2 Show that'C, = °C, 3° How many different ways are there of selecting 2 3 photographs from 10 photographs b 5 books from 7 books € team of 11 footballers from 14 footballers? 4 How many different combinations of 3 letters can be chosen from the leters PQRS.T? 5 The diagram shows 2 different boxes, A and B. 8 different toys are to be placed in the boxes. “ 3 Find the number of ways in which the 8 toys ean ‘be placed in the boxes so that 5 toys are in ‘box A and 3 toys are in box B. 6 4 pencils and 3 pens are to be selected from a collection of 8 pencils and S pens. Find the number of different selections that can be made. 7 Four of the letters of the word PAINTBRUSH are selected at random. Find the number of different combinations if 2 there is no restriction om the letters selected b the letter T must be selected & A test consists of 30 questions, Each answer is cither correct or incorrect. Find the number of different ways in which itis possible to answer a exactly 10 questions correctly b exactly 25 questions correctly. 243 >> CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK 9 Anathletics club has 10 long distance runners, 8 sprinters and 5 jumpers ‘A team of 3 long distance runners, 5 sprinters and 2 jumpers is to be selected, Find the number of ways in which the team ean be selected, 10. A team of 5 members is to be chosen from 5 men and 3 women. Find the number of different teams that can be chosen @ if there are no restrictions b that consist of 3 men and 2 women © that consist of no more than 1 woman, 11. A committee of 5 people is to be chosen from 6 women and 7 men. Find the number of different committees that can be chosen. a if there are no restrictions b if there are more men than women. 12 A.committee of 6 people is to be chosen from 6 men and 7 women. ‘The committee must contain at least |-man. Find the number of different committees that can be formed. 13. A school committee of 5 people is to be selected from a group of 4 teachers, and 7 students. Find the number of different ways that the committee can be selected if a there are no restrictions b there must be atleast | teacher and there must be more students than teachers. 14 A west consists of 10 different questions. 4 of the questions are on trigonometry and 6 questions are on algebra, Students are asked to answer 8 questions. 2 Find the number of ways in which students can select 8 questions if there are no restrictions, b Find the number of these selections which contain at least 4 algebra questions, 15. Rafiu has downloaded 10 movies to his smartphone, 4 of the movies are comedies, 3 are thrillers and 3 are science fiction. He is going to select 5 of these movies. Find the number of ways he can make his selection if 2 there are no restrictions b his selection must contain his favourite thriller movie. ¢ his selection must contain at least 3 comedy movies. 16 Ina group of 15 entertainers, there are 6 singers, 5 guitarists and 4 comedians. A show is to be given by 6 of these entertainers. In the show, there must be atleast 1 guitarist and I comedian, ‘There must also be more singers than guitarist, Find the number of ways that the 6 entertainers can be selected, 17. Given that 45 * "Cy= (n+ 1) x"'Cs, find the value of 1. Ss 244)10 Permutations and combinations In this chapter you have learned about permutations and combinations. Explain to a friend the difference hetween a permutation and a combination. Ey Tio ‘The number of ways of arranging n distinct items in a Tine is nx (n~I) X(N 2)X... X3X 2X1 Sl ‘The number of permutations of + items from 1 distinet items is nt "p, f= In permutations, order matters. ‘The number of combinations of r items from n distinct items is mn) 0) ria In combinations, order does not matter. Past paper questions Worked example 4a i Find how many different 4-digit numbers can be formed using the digits 1, 2, 3,4, 5 and 6 if ‘no digit is repeated. nw How many of the 4-digit numbers found in part i are greater than 6000? w How many of the 4-digit numbers found in part i are greater than 6000 and are odd? a b A quiz team of 10 players is to be chosen from a class of 8 boys and 12 girls. i Find the number of different teams that can be chosen if the team has to have equal numbers of agitls and boys. BI fi Find the number of different teams that can be chosen if the team has 10 include the youngest and oldest boy and the youngest and oldest girl, Py Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q10 Nov 2014 Answers 1a 1 Number of 4-digit numbers =“P, = 360 ii Method 1 ‘The first number must be a 6. 24s >> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK ‘There are now three spaces to fill using three of the remaining five digits 1,2,3,4and5, st__ st ‘Number of ways of filling the remaining three spaces = *P, 6 There are 60 different numbers that satisfy the conditions. Method 2 ‘Consider the number of choices for filling each of the four spaces. i 5 4 3 choice choices choices choices Number of ways of filling the four spaces = 1 x $4 x 3, ‘There are 60 different numbers that satisfy the conditions. ‘Method 1 The first number must be a 6, ‘The last number must be a 1, 3 or 5. ‘The middle two spaces must then be filled using two of the remaining four numbers, Number of ways of filling the four spaces =1 “P, x 3 = 1x 36 There are 36 diferent numbers that satisfy the conditions. Method 2 ‘Consider the number of choices for filling each of the four spaces, The first number must be a 6. ‘The last number must be a 1, 3 or 5. ‘When the first and last spaces have been filled there will be four numbers left to choose from, Eis eet choice choices choices choices Number of ways of filling the four spaces = 1 x43 3. There are 36 different numbers that satisfy the conditions. b i There must be 5 boys and 5 girls, ‘Number of ways of choosing 5 boys from 8 = °C; Number of ways of choosing 5 girls from 12 = "°C. Number of possible teams = °C; x "C; = 56 x 792 = 44 352 ii The team includes the youngest and oldest boy and the youngest and oldest girl There are now 6 places left to fill and 16 people left to choose from. Number of ways of choosing 6 from 16 = "°C, = 8008 Number of possible teams = 8008 SESE See) 246 >10 Permutations and combinations 41 @ How many even numbers Jess than 500 can be formed using the digits 1, 2, 3, and 5? Each digit may be used only once in any number, i] b Acommittee of 8 people is to be chosen from 7 men and 5 women, Find the number of diferent committees that could be selected if i the committee contains at least 3 men and at least 3 women, (1 lithe oldest man or the oldest woman, but not both, must be included in the committee, fel Cambridge IGCSE Additional Mathemetics 0606 Paper 11 Q10 Jun 2014 2 a Jean has nine different flags. i Find the number of different ways in which Jean ean choose three flags from her nine flags. ty Ji Jean has five agpoles jn a sow: She purty one of her aine Rags on each flagpote. Calculate the number of different five-flag arrangements she can make, 0 b The six digits of the number 738925 are rearranged so that the resulting six-digit number is even. Find the number of different ways in which this ean be done. el Cambridge IGCSE Additional Mathematics 0606 Paper 22 Q2 Mar 2015 3 a A lock can be opened using only the number 4351. State Whether this is a permutation or a combination of digits, giving a reason for your answer. i b There are twenty numbered balls in a bag. Two of the balls are numbered 0, six are numbered 1, five are numbered 2 and seven are numbered 3, as shown in the table below. oti f2f3 Ee en Four of these balls are chosen at random, without replacement, Calculate the number of ways this ean be done so that ithe four balls all have the same number, p {ithe four balls all have different numbers, 2 ‘ik the Rour balls have numbers that total 3 8 Cambridge IGCSE Additional Mathematics 0606 Paper 21. OS Jun 2015 4 a 6 books are to be chosen at random from 8 different books. i Find the number of different selections of 6 books that could be made. aw {A clock is to be displayed on a shelf with 3 of the 8 different books on each side of it. Find the number of ways this can be done if there are ao restrictions on the choice of books, mw 3 of the 8 books are music books which have to be kept together. 2 b A team of 6 tennis players is to be chosen from 10 tennis players consisting of 7 men and 3 women. Find the number of differea teants that could be efiosen if the team must inchide at least | woman. 8] Cambridge IGCSE Additional Mathematics 0606 Paper 11 4 Nov 2015 5 2 A 6-character password is to be chosen from the following 9 characters, letters A B E F numbers Sk symbols LE Bach character may be used only once in any password. Find the number of different 6-character passwords that may be chosen if aa SSS 207 >> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK i there are no restrictions, 0 ji the password must consist of 2 letters, 2 numbers and 2 symbols, in that order, PI Iii the password must start and finish with @ symbol. 2) 'b Amexamination consists of a section A, containing 10 short questions, and a section B, containing $ long questions. Candidates are required to answer 6 questions from section A and 3 questions from section B Find the number of different selections of questions that ean be made if i there are no further restrictions 2 li candidates must answer the first 2 questions in section A and the first question in section B. pI Cambridge 1GCSE Additional Mathematies 0606 Paper 12 05 Mar 2016 6 a The letters of the word THURSDAY are arranged in a straight line, Find the number of different ‘arrangements of these letters if there are no restrictions, a ‘the arrangement must start with the letter T and end with the letter Y, a iii the second letter in the arrangement must be Y. it b 7 children have to be divided into two groups, one of 4 children and the other of 3 children, Given that there are 3 girls and 4 boys, find the number of different ways this can be done if i there are no restrictions, uw) all the boys are in one group, i ‘one boy and one gir] are twins and must be in the same group. By) Cambridge IGCSE Additional Mathematics 0606 Paper 12 6 Mar 2017 7 a A football club has 30 players. In how many different ways can a captain and a vice-captain be selected at random from these players? ty b A team of 11 teachers isto be chosen from 2 mathematics teachers, 5 computing teachers and 9 science teachers. Find the number of different teams that can be chosen if the team must have exactly 1 mathematics teacher, 2 {ithe team must have exactly 1 mathematics teacher and at least 4 computing teachers, il Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q8 Jun 2017 8 A group of five people consisis of two women, Alice and Betty, and three men, Carl, David and Ed. 1 Three of these five people are chosen at random to be a chairperson, a treasurer and a secretary, Find the number of ways in which this can be done if the chairperson and treasurer are both men. Pl ‘These five people sit in a row of five chairs Find the number of different possible seating arrangements if David must sit in the middle, ii} ii Alice and Carl must sit together. 2) Cambridge IGCSE Additional Mathematics 0606 Paper 22 Q3 Mar 2018 9 A 7-character password isto be selected from the 12 characters shown in the table. Each character may be used only once. A B io D e t zg h D 2 3 410 Permutations and combinations Find the number of different passwords, if there are no restrictions, uw that start with a digit, a that contain 4 upper-case letters and 3 lower-case letters such that all the upper-case letters are together and aii the fower-case letters are together. a Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q3 Jun 2018 10 A 5.digit code is to be formed from the digits 1, 2, 3,4, 5, 6, 7, 8, 9. Each digit can be used once only it any code. Find how many codes ean be formed if i the first digit of the code is 6 and the other four digits are odd, 2 ji each of the first three digits is even, 2 iii the first and fast digits are prime. 2 Cambridge IGCSE Additional Mathematics 0606 Paper 21 06 Nov 2018 a i Find how many 4-digit numbers ean be formed using the digits 1, 3, 4, 6, 7 and 9, Each digit may be used once only in any 4-digit number. tw i How many of these 4-digit mumbers are even and greater than 60002 8 A committee of 5 people is to be formed from 6 doctors, 4 dentists and 3 nurses. Find the number of different committees that could be formed if i there are no restrictions, 0 fi the committee contains at least one doctor, @ iii the committee contains all the nurses, uy Cambridge IGCSE Additional Mathematies 0606 Paper 11 Q5 Nov 2020 249)