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- Question : EX1.1 - If there are 10 ways of doing A, 3 ways of doing B and 19 ways of doing C, how many ways are there of doing (a) (i) both A and B? (ii) both B and C? (b) (i) either A or B? (ii) either A or C?
- Question : EX1.2 - If there are 4 ways of doing A, 7 ways of doing B and 5 ways of doing C, how many ways are there of doing (a) all of A, B and C? (b) exactly one of A, B or C?
- Question : EX1.3 - How many diff erent paths are there (a) from A to C? (b) from C to E? (c) from A to E?
- Question : EX1.4 - Jamil is planting out his garden and needs one new rose bush and some dahlias.Th ere are 12 types of rose and 4 varieties of dahlia in his local nursery. How many possible selections does he have to choose from?
- Question : EX1.5 - A lunchtime menu at a restaurant off ers 5 starters, 6 main courses and 3 desserts. How many diff erent choices of meal can you make if you would like (a) a starter, a main course and a dessert? (b) a main course and either a starter or a dessert? (c) any two diff erent courses?
- Question : EX1.6 - Five men and three women would like to represent their club in a tennis tournament. In how many ways can one mixed doubles pair be chosen?
- Question : EX1.7 - A mathematics team consists of one student from each of years 7, 8, 9 and 10. Th ere are 58 students in year 7, 68 in year 8, 61 in year 9 and 65 in year 10. (a) H ow many ways are there of picking the team? Y ear 10 is split into three classes: 10A (21 students), 10B (23 students) and 10C (21 students). (b) I f students from 10B cannot participate in the competition, how many ways are there of picking the team?
- Question : EX1.8 - Student passwords consist of three letters chosen from A to Z, followed by four digits chosen from 1
- Question : EX1.9 - A beetle walks along the edges from the base to the tip of an octahedral sculpture, visiting exactly two of the middle vertices (A, B, C or D). How many possible routes are there?
- Question : EX1.10 - Professor Square has 15 diff erent ties (seven blue, three red and fi ve green), four waistcoats (red, black, blue and brown) and 12 diff erent shirts (three each of red, pink, white and blue). He always wears a shirt, a tie and a waistcoat. (a) H ow many diff erent outfi ts can he make? Professor Square never wears any outfi t that combines red with pink. (b) H ow many diff erent outfi ts can he make with this limitation?
- Question : EX1.11 - How many diff erent three digit numbers can be formed using the digits 1,2,3,5,7 (a) o nce on ly? (b) if digits can be repeated?
- Question : EX1.12 - In how many ways can (a) four toys be put into three boxes? (b) three toys be put into fi ve boxes?
- Question : EX2.1 - Evaluate: (a) (i) 5 ! (ii) 6 ! (b) (i) 2
- Question : EX2.2 - Evaluate: (a) (i) 8 ! (ii) 1 1! (b) (i) 9
- Question : EX2.3 - Find the number of ways of arranging: (a) 6 ob jects (b) 8 ob jects (c) 2 6 ob jects.
- Question : EX2.4 - (a) H ow many ways are there of arranging seven textbooks on a shelf? (b) I n how many of those arrangements is the single biggest textbook not at either end? [5 marks]
- Question : EX2.5 - (a) H ow many fi ve-digit numbers can be formed by using each of the digits 1
- Question : EX2.6 - A class of 16 pupils and their teacher are queuing outside a cinema. (a) How many diff erent arrangements are there? (b) H ow many diff erent arrangements are there if the teacher has to stand at the front?
- Question : EX2.7 - A group of nine pupils (fi ve boys and four girls) are lining up for a photograph, with all the girls in the front row and all the boys at the back. How many diff erent arrangements are there?
- Question : EX2.8 - (a) H ow many six-digit numbers can be made by using each of the digits 1
- Question : EX2.9 - A class of 30 pupils are lining up in three rows of ten for a class photograph. How many diff erent arrangements are possible?
- Question : EX2.10 - A baby has nine diff erent toy animals. Five of them are red and four of them are blue. She arranges them in a line so that the colours are arranged symmetrically. How many diff erent arrangements are possible?
- Question : EX3.1 - Fully simplify the following fractions:
- Question : EX3.2 - Write these as a ratio of 2 factorials: (a) (i) 7
- Question : EX3.3 - Explain why 6!
- Question : EX3.4 - Solve n! (n )! = 20 where n is a positive integer.
- Question : EX3.5 - Solve ( )! ( )! = 990.
- Question : EX3.6 - Solve n ! (nn ?1))!! = 16(((nn )! for n ?N.
- Question : EX4.1 - Evaluate:
- Question : EX4.2 - (a) (i) In how many ways can six objects be selected from eight? (ii) I n how many ways can fi ve objects be selected from nine? (b) (i) In how many ways can either three objects be selected from ten, or seven objects be selected from 12? (ii) In how many ways can either two objects be selected from fi ve, or three objects be selected from four? (c) (i) I n how many ways can fi ve objects be selected from seven and three diff erent objects be selected from eight? (ii) In how many ways can six objects be selected from eight and three diff erent objects be selected from seven?
- Question : EX4.3 - Solve:
- Question : EX4.4 - An exam paper consists of 15 questions. Students can select any nine questions to answer. How many diff erent selections can be made?
- Question : EX4.5 - Suppose you are revising for seven subjects, but you can only complete three in one evening. (a) In how many ways can you select three subjects to do on Monday evening? (b) I f you have to revise mathematics on Tuesday, in how many ways can you select the subjects to do on Tuesday evening?
- Question : EX4.6 - In the
- Question : EX4.7 - Th ere are 16 boys and 12 girls in a class. Th ree boys and two girls are needed to take part in the school play. In how many diff erent ways can they be selected?
- Question : EX4.8 - A soccer team consists of one goalkeeper, four defenders, four midfi elders and two forwards. A manager has three goalkeepers, eight defenders, six midfi elders and fi ve forwards in the squad. In how many ways can she pick the team?
- Question : EX4.9 - A school is planning some trips over the summer. Th ere are 12 places on the Greece trip, ten places on the China trip and ten places on the Disneyland trip. Th ere are 140 pupils in the school who are all happy to go on any of the three trips. In how many ways can the spaces be allocated?
- Question : EX4.10 - A committee of three boys and three girls is to be selected from a class of 14 boys and 17 girls. In how many ways can the committee be selected if: (a) Ana has to be on the committee? (b) t he girls must include either Roberta or Priya, but not both?
- Question : EX4.11 - Raspal
- Question : EX4.12 - An English examination has two sections. Section A has fi ve questions and section B has four questions. Four questions must be answered in total. (a) H ow many diff erent ways are there of selecting four questions to answer if there are no restrictions? (b) H ow many diff erent ways are there of selecting four questions if there must be at least one question answered from each section?
- Question : EX4.13 - 15 points lie around a circle. Each point is connected to every other point by a straight line. How many lines are formed in this way?
- Question : EX4.14 - Ten points are drawn on a sheet of paper so that no three lie in a straight line. By connecting up the points: (a) how many diff erent triangles can be drawn? (b) how many diff erent quadrilaterals can be drawn?
- Question : EX4.15 - At a party, everyone shakes hands with everyone else. If 276 handshakes are exchanged, how many people are there at the party?
- Question : EX4.16 - A group of 45 students are to be seated in three rows of 15 for a school photograph. Within each row, students must sit in alphabetical order according to name, but there is no restriction determining the row in which a student must sit. How many diff erent seating arrangements are possible, assuming no students have identical names?
- Question : EX5.1 - How many of the numbers between 101 and 800 inclusive are not divisible by 5?
- Question : EX5.2 - How many permutations of the letters of the word JUMPER do not start with a J?
- Question : EX5.3 - A bag contains 12 diff erent chocolates, four diff erent mints and six diff erent toff ees. Th ree sweets are chosen. How many ways are there of choosing (a) all not chocolates? (b) not all chocolates?
- Question : EX5.4 - How many permutations of the letters of KITCHEN (a) do not begin with KI? (b) do not have K and I in the fi rst two letters?
- Question : EX5.5 - A committee of fi ve people is to be selected from a class of 12 boys and nine girls. How many such committees include at least one girl?
- Question : EX5.6 - In a word game there are 26 letter tiles, each with a diff erent letter. How many ways are there of choosing seven tiles so that at least two are vowels?
- Question : EX5.7 - A committee of six is to be selected from a group of ten men and 12 women. In how many ways can the committee be chosen if it has to contain at least two men and one woman?
- Question : EX5.8 - Seven numbers are chosen from the integers 1?19 inclusive. How many have (a) at most two even numbers? (b) at least two even numbers?
- Question : EX5.9 - How many permutations of the letters DANIEL do not begin with D or do not end with L? [6 marks]
- Question : EX6.1 - Evaluate:
- Question : EX6.2 - Find the number of permutations of: (i) 4 objects out of 10 (ii) 6 objects out of 7.
- Question : EX6.3 - Find the number of ways of selecting when order matters: (i) 3 objects out of 5 (ii) 2 objects out of 15.
- Question : EX6.4 - Solve the following equations:
- Question : EX6.5 - In a
- Question : EX6.6 - A teacher needs to select four pupils from a class of 24 to receive four diff erent prizes. How many possible ways are there to award the prizes?
- Question : EX6.7 - How many three-digit numbers can be formed from digits 1
- Question : EX6.8 - Eight athletes are running a race. In how many diff erent ways can the fi rst three places be fi lled?
- Question : EX6.9 - An identifi cation number consists of two letters followed by four digits chosen from 0?9. No digit or letter may appear more than once. How many diff erent identifi cation numbers can be made?
- Question : EX6.10 - Prove that Pnn?1 nPnn.
- Question : EX6.11 - Th ree letters are chosen from the word PICTURE and arranged in order. How many of the possible permutations contain at least one vowel?
- Question : EX6.12 - Eight runners compete in a race. In how many diff erent ways can the three medals be awarded if James wins either a gold or a silver?
- Question : EX6.13 - A class of 18 needs to select a committee consisting of a President, a Secretary and a Treasurer. How many diff erent ways are there to form the committee if Rajid does not want to be President?
- Question : EX6.14 - Solve the equation 2nP33 66Pnn, justifying that you have found all solutions.
- Question : EX7.1 - In how many ways can 14 people be arranged in a line if Joshua and Jolene have to stand together?
- Question : EX7.2 - Students from three diff erent classes are standing in the lunch queue. Th ere are six students from 10A, four from 10B and four from 10C. In how many ways can the queue be arranged if students from the same class have to stand together? [
- Question : EX7.3 - In how many ways can six Biology books and three Physics books be arranged on the shelf if the three Physics books are always next to each other?
- Question : EX7.4 - In a photo there are three families (six Greens, four Browns and seven Grays) arranged in a row. Th e Browns have had an argument so no Brown will stand next to another Brown. How many diff erent permutations are permitted?
- Question : EX7.5 - In a cinema there are 15 seats in a row. In how many ways can seven friends be seated in the same row if (a) there are no restrictions? (b) they all want to sit together?
- Question : EX7.6 - Five women and four men stand in a line. (a) I n how many arrangements will all four men stand next to each other? (b) I n how many arrangements will all the men stand next to each other and all the women stand next to each other? (c) In how many arrangements will all the men be apart? (d) In how many arrangements will all the men be apart and all the women be apart?

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