Chapter 34 Permutation Combination 10202021 Permutation and combination

Chapter 34 © Permutation & Combination 排列与�合 10/20/2021 Permutation and combination by ® Chtan, FYHS-Kulai © 1

Permutation and Combination are the fundamental knowledges prior to study the Probability and Statistics. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 2

Concepts : How to differentiate Permutation and Combination ? 排列与组合的区别? 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 3

Let us examine the following examples : Q: 4 basketball teams A, B, C and D take part in a competition, each team must meet each other once. (i) What are the possible results of the first two positions? (ii) How many matches in this competition? 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 4

Soln: (i) 10/20/2021 Champion Runner up A A B C A D B B C D C D D D A A A B B C There are 12 possibilities. Permutation and combination by Chtan, FYHS-Kulai 5

(ii) A A A B B C D C D D The competition has six matches. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 6

From the above solution, AB and BA are two different arrangements (permutation). They form a combination. This means Permutation requires the sequence or order of the objects to do the arrangement. Combination no need to consider the order of the objects. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 7

Let us see another example : Two cards are drawn from a deck of 52 cards. In Permutation, the sequence of the appearance of “Club A” is matter. There are 52 X 51 ways of appearances. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 8

In Combination, These are the same, so there are (52 X 51)/2 ways. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 9

Addition theorem and multiplication theorem. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 10

e. g. 1 Everyday from KL to Singapore, there are 2 shifts of railways, 4 coaches and 3 flights. How many ways from KL to Singapore daily? 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 11

Soln : There are 2+4+3=9 ways. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 12

Addition theorem In general, there are r independent methods to complete a job. Each method has different number of ways to do. Say, method 1 has n 1 ways, method 2 has n 2 ways, method 3 has n 3 ways … method r has nr ways. Hence, total number of ways=n 1+n 2+n 3+…+nr 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 13

e. g. 2 There are 2 ways from point A to point B, 3 ways from point B to point C. how many ways from A to C? Soln : 2 X 3 = 6 ways 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 14

Multiplication theorem There are r steps to complete an event. Step 1 has n 1 ways, step 2 has n 2 ways, step 3 has n 3 ways, …, step r has nr ways. The number of ways to complete this job is n 1 xn 2 xn 3 x…xnr 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 15

e. g. 3 There are 3 towns A, B and C in Wonderland. Six roads go from A to B, and 4 roads go from B to C. In how many ways can one drive from A to C? B A 10/20/2021 C 6 x 4 = 24 ways Permutation and combination by Chtan, FYHS-Kulai 16

A new town called D and several new roads were built in Wonderland (see figure). How many ways are there to drive from A to C now? D B C A 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 17

Soln : 6 x 4 + 3 x 2 = 24 + 6 ways Multiplication Thm and Addition Thm 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 18

Permutation (arrangements) 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 19

The number of ways of arranging n unlike objects in a line is n! Note : n!=n(n-1)(n-2)(n-3)…(3)(2)(1) 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 20

e. g. 4 How many different number plates can be formed if each is to contain the three letters A, C and E followed by the three digits 4, 7, 8? Soln: There are 3! ways of arranging the letters A, C and E. There also 3! ways of arranging the digits 4, 7 and 8. Hence, total number of different plates=(3!)=36 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 21

If there are n different objects, r objects taken from the objects to arrange in a line. We have : Position : 1 2 3 … r Number of ways : By multiplication rule, we got : 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai ways 22

This is denoted by or or Note : read as “P”, “n”, “r” 10/20/2021 !!!!! Permutation and combination by Chtan, FYHS-Kulai 23

Hence, Note : read as “P”, “n”, “r” 10/20/2021 !!!!! Permutation and combination by Chtan, FYHS-Kulai 24

e. g. 5 Find the value =10 x 9 x 8 + 8 x 7 x 6 x 5 =720+1680 =2400 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 25

e. g. 6 Three letters are taken from the word “VOLUME” to arrange as a line. How many ways to do? Soln: 10/20/2021 There are 6 letters. Permutation and combination by Chtan, FYHS-Kulai 26

e. g. 7 4 unlike digits and 4 different letters are to be arranged in a line so that both ends are letters. Soln: 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 27

There are ways that both ends are letters. And ways for the rest of the 6 letters in between. Hence, total number of ways = x = 4 x 3 x 6! = 8640 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 28

e. g. 8 7 digits 1, 2, 3, 4, 5, 6, 7. if each digit can be used once, find a) How many 4 digits can be formed ? b) How many 4 -digit odd numbers ? 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 29

Soln: (a) =7 x 6 x 5 x 4 = 840 ways (b) 6 Hence, 10/20/2021 5 4 X odd number (1, 3, 5, 7), 4 numbers 4=6 x 5 x 4 x 4 = 480 ways Permutation and combination by Chtan, FYHS-Kulai 30

e. g. 9 Prove that 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 31

Soln: = 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 32

e. g. 10 How many odd number between 4000 and 9000 if each digit is different from one to another. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 33

10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 34

Soln: Case 1 : 3 8 7 5 (1), (3), (5) (7), (9) (4), (6), (8) 3 x 8 x 7 x 5=840 ways 2 Case 2 : 8 7 (1), (3), (5/7) , (9) (5), (7) 2 x 8 x 7 x 4=448 ways 10/20/2021 4 Total=840+448=1288 Permutation and combination by Chtan, FYHS-Kulai 35

e. g. 11 From 0, 1, 2, 3, 4, 5 six numbers, no repetition is allowed, how many ways to form (a) a 3 -digit number (b) a 4 -digit even number (c) a 3 -digit number which is divisible by 5. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 36

e. g. 12 5 boys and 5 girls are to be arranged in a line, how many arrangements if (a) 2 identified girls must be next to each other, (b) if the 2 girls are not next to each other? 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 37

Soln: (a) G 1 G 2 Consider G 1 and G 2 as 1 object. Altogether 9 objects. Hence, 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 and then G 1 and G 2 has 2 permutations! Total number =9!x 2!=725760 ways 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 38

e. g. 13 All letters in the word “TRIANGLE” are taken to arrange in a line. How many arrangements are there if the two ends contain no “T” or “E”? (Ans: 21600) 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 39

Coloring Maps 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 40

Q : Use only 5 different colors to color the following different maps if colors can be reuse. The adjacent regions cannot have the same color. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 41

Case 1: A B C D 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 42

Case 2: A E D C B 10/20/2021 A Permutation and combination by Chtan, FYHS-Kulai 43

Case 3: E A F B 10/20/2021 C Permutation and combination by Chtan, FYHS-Kulai D G 44

Case 4: E F 10/20/2021 D A B C Permutation and combination by Chtan, FYHS-Kulai G 45

Case 5: B 10/20/2021 C A D E Permutation and combination by Chtan, FYHS-Kulai 46

Case 6: C F B E A D 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 47

Case 7: D E A C B F 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 48

e. g. Starting from A, cannot go backward, a routine is ended when the same point is reached the 2 nd time. A B 10/20/2021 D Permutation and combination by Chtan, FYHS-Kulai C 49

e. g. Draw n different lines in a plane, find the maximum regions that can be divided by them. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 50

Repetitive elements in Permutation and combination by Chtan, FYHS-Kulai 10/20/2021 51

The number of ways of arranging in a line n objects, of which p are alike, is : 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 52

In general, the number of ways of arranging in a line n objects, of which p are alike, q are alike, r are alike is : 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 53

e. g. 14 Consider the letters A, B, C, D There are 4!=4 x 3 x 2 x 1 ways to arrange these 4 letters. For example, if, instead of A, B, C, D we have the letters A, A, A, D. So, now, we have 4!/3! =4 ways. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 54

e. g. 15 In how many ways can the letters of word STATISTICS be arranged? Soln: There are 10 letters, S occurs 3 times, T occurs 3 times, I occurs twice. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 55

e. g. 16 Find the permutation for the word “MISSISSIPPI”. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 56

e. g. 17 Three boys and three girls are to be seated in a row. Calculate the total number of arrangements if all the three boys want to sit together and all three girls also want to sit together. (ans : 72) 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 57

e. g. 18 A man climbs the stairs as shown. He can climb with a single step or two steps each time. How many possible ways he can climb the stairs? (Ans : 144) 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 58

10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 59

1 step 2 steps No. of ways 11 0 11!/11!=1 9 7 5 3 1 1 2 3 4 5 10!/9!=10 9!/(7!2!)=36 8!/(5!3!)=56 7!/(3!4!)=35 6!/(1!5!)=6 Total = 144 ways 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 60

e. g. 19 How many routes? How many routes are possible from point A to point B if you always move toward the target? A B 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 61

e. g. 20 In how many ways can 3 letters to be dropped in 6 mailing boxes? 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 62

e. g. 21 In how many ways can 6 letters to be dropped in 3 mailing boxes? 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 63

Q: How many ways can 4 one and 3 two be arranged so that no 2 will be next to each other. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 64

10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 65

10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 66

10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 67

Circular Permutation 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 68

Case 1 The number of ways of arranging n unlike objects in a ring, when clockwise and anticlockwise arrangements are different, is 10/20/2021 (n-1)! Permutation and combination by Chtan, FYHS-Kulai 69

Consider 4 people A, B, C and D, who are to be seated at a round table. The following 4 arrangements are the same : A D D BC C 10/20/2021 AB B B C DA A Permutation and combination by Chtan, FYHS-Kulai C D 70

Therefore the number of arrangements of 4 people around the table is 3!=6. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 71

Case 2 The number of ways of arranging n unlike objects in a ring, when clockwise and anticlockwise arrangements are the same , is 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 72

For example, if A, B, C and D are 4 different coloured beads which are threaded on a ring, then the following two arrangements are the same – the one is the other viewed from the other side. A D B C 10/20/2021 A B D C Permutation and combination by Chtan, FYHS-Kulai 73

A D D A B B C C A D B A 10/20/2021 A B D C A D B C D A A B C C D B B D A A C C B D D C A D A Permutation and combination by Chtan, FYHS-Kulai B C 74

Therefore the number of arrangements of 4 beads on a ring is 3!/2 = 3. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 75

e. g. 22 Nine children play a party game and hold hands in a circle. In how many different ways can this be done? 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 76

e. g. 23 One white, one blue, one red and two yellow beads are threaded on a ring to make a bracelet. Find the number of ways of arranging the beads on a ring. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 77

Soln: If all the objects are unlike, the number of ways of arranging five beads on a ring is (5 -1)!/2 = 4!/2 There are 2 yellows, hence, 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 78

e. g. 24 6 men and 6 women are to be seated on a round table. How many ways are there if no two men are allowed to seat next to each other? (ans: 86400) 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 79

e. g. 25 40 different beads are used to make a ring. How many ways to do? If 20 beads are selected to thread as a ring, how many ways? 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 80

Same elements to be re-used in Permutation 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 81

e. g. 26 Use the five digits 2, 3, 4, 5, 9 to obtain a permutation of a number which must greater than 5000 and less than 10000. If repetition is allowed, how many ways to do ? Soln: 2 5 5 5 =2 x 5 x 5 x 5=250 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 82

e. g. 27 We call a natural number “odd-looking” if all of its digits are odd. How many four-digit odd-looking numbers are there? Soln: For each digit, there are 5 possible numbers to be filled (1, 3, 5, 7, 9). So, there are 5 x 5 x 5 = 625 numbers. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 83

e. g. 28 We toss a coin three times. How many different sequences of heads and tails can we obtain? Soln: 1 st time 2 nd time 3 rd time Possible outcomes = 2 x 2 = 8. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 84

e. g. 29 Each box in a 2 x 2 table can be colored black or white. How many different colorings of the table are there? Soln: Each box can only be filled either black or white. So, 4 boxes to be colored. i. e. 2 x 2 x 2 = 16 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 85

e. g. 30 How many ways are there to fill in a Special Sport Lotto card? In this lotto you must predict the results of 13 hockey games, indicating either a victory for one of two teams, or a draw. Soln: 3 x 3 x 3 x …x 3 = 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 86

e. g. 31 The Hermetian alphabet consists of only 3 letters : A, B and C. A word in this language is an arbitrary sequence of no more than four letters. How many words does the Hermetian language contain? Soln: The language can have word with 1 -letter, 2 letter, 3 -letter or 4 -letter words. Hence, there are 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 87

e. g. 32 Six different books are to be distributed to 3 students, (a) if each student can have only one book, how many ways can be done? (b) if all six books are to be given away, no restriction is imposed, how many ways can it be distributed? (Ans : a) 120 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai b) 729) 88

成功是差一点点失败， 失败是差一点点成功 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 89

Combination �合 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 90

Notation : or 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 91

How to calculate the number of ways from taking r objects from n objects ? Let us consider 4 different elements a, b, c, d. Say, we only take 3 out of the 4 elements. The relation between Permutation and Combination : 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 92

Combination Permutation a b d abc acb abd adb a c d acd cad dac adc cda dca b c d bcd cbd dbc bdc cdb dcb a b c 10/20/2021 bac bca bad bda Permutation and combination by Chtan, FYHS-Kulai cab cba dab dba 93

From the above table, each combination has six different permutations. Hence, the permutation of 3 elements taken from 4 elements can be calculated in the following steps: Step 1 : Step 2 : Each combination has By multiplication rule : 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 94

In general, 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 95

Note : 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 96

e. g. 33 Calculate Soln: 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 97

e. g. 34 8 persons were chosen from a group of 50 people. How many ways of this selection? Soln: 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 98

e. g. 35 A company want to recruit 6 salesgirls and 4 sales assistant. If there are 9 applicants for the salesgirls and 6 applicants for the sales assistant, how many ways of this selection? Soln: 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 99

e. g. 36 Prove that 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 100

Soln: 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 101

e. g. 37 Ten points are marked on a plane so that no three of them are on the same straight line. How many triangles are there with vertices at these points? Soln: 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 102

e. g. 38 A special squad consists of 3 officers, 6 sergeants, and 60 privates. In How many ways can a group consisting of 1 officer, 2 sergeants, and 20 privates be chosen for an assignment? Soln: 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 103

e. g. 39 Ten points are marked on a straight line, and 11 points are marked on another line, parallel to the first one. How many (a) triangles; (b) quadrilaterals are there with vertices at these points? 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 104

e. g. 40 A set of 15 different words is given. In how many ways is it possible to choose a subset of no more than 5 words? Soln: 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 105

Some properties of the Combinations: (1) 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 106

Proof : We have, 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 107

e. g. 41 Solve the equation : Soln: Obviously, we can’t have x=x-8. But, 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 108

(2) * 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 109

Proof : 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 110

e. g. 42 Prove that : Proof : 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 111

(3) 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 112

Proof : Recall 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 113

Put x=1, a=1 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 114

Miscellaneous Examples Permutation and combination by Chtan, FYHS-Kulai 10/20/2021 115

e. g. 43 A captain and a deputy captain must be elected in a soccer team with 11 players. How many ways are there to do this? Soln: We have 11 x 10 = 110 different outcomes 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 116

e. g. 44 How many ways are there to put one white and one black rook on a chessboard so that they do not attack each other? Soln: 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 117

10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 118

The black (or white) rook can attack 15 positions from it stands. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 119

So, there left 64 -15=49 positions for the second rook to put! Hence, there are 64 x 49 =3136 ways. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 120

e. g. 45 How many ways are there to put one white and one black king on a chessboard so that they do not attack each other? Soln: 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 121

Case 1 Only consider when the white king is placed at the corner. 4 X 60 Only 4 corners 10/20/2021 Each corner, white king can move 4 possible steps. So, black king can be placed on the remaining 60 positions. Permutation and combination by Chtan, FYHS-Kulai 122

Case 2 24 X 58 24 possible spaces for the white king to be placed! The white king has 6 possible moves in each place. The remaining 64 -6=58 possible positions is for the black king! 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 123

Case 3 The white king is placed in the centre region. There are 36 such positions. 36 x 55 Each position for the white king has 9 possible moves. So, the 64 -9=55 remaining positions for the black king! 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 124

Hence, there are = 4 x 60 + 24 x 58 + 36 x 55 = 3612 ways. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 125

e. g. 46 The map of a town is depicted in the following figure. All its streets are one-way, so that you can drive only “east” or “north”. How many different ways are there to reach B starting from A? B 10/20/2021 A Permutation and combination by Chtan, FYHS-Kulai 126

Soln: or 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 127

e. g. 47 Six boxes are numbered 1 through 6. How many ways are there to put 20 identical balls into these boxes so that none of them is empty? 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 128

Soln: Box 1 Box 2 Box 3 Box 4 11 Box 5 13 15 Box 6 17 19 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 There are 20 -1=19 gaps between 20 balls. Any wall can be in any of these 19 gaps. No two of the wall can be in the same gap. Therefore, the number of all possible partitions is. 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai 129

TYPES FORMULA arranging n unlike objects in a line Arranging n objects of which p of one type are alike, q of another type are alike, r of a third type are alike, and so on Arranging n unlike objects in a ring when clockwise and anticlockwise arrangements are different Arranging n unlike objects in a ring when clockwise and anticlockwise arrangements are the same 10/20/2021 To be continue… Permutation and combination by Chtan, FYHS-Kulai 130

Continue… TYPES The number of permutation of r objects taken from n unlike objects The number of combination of r objects taken from n unlike objects 10/20/2021 Permutation and combination by Chtan, FYHS-Kulai FORMULA 131

The 2009 year end examination scopes : 1. The straight line 2. The circle 3. The parabola 4. The ellipse 5. The hyperbola 排列与组合 6. Permutations & combinations 7. Probability概率 10/20/2021 hyperbola 132

nd 2 November, 2009 (Monday) S 2 S Mathematics Final Examination Part A : Short Questions -- answer all 9 questions x 5%=45% Part B : Long Questions -- answer 5 from 8 questions x 11%=55% 10/20/2021 hyperbola 133