Learning Outcomes: i. Use the techniques of counting. [Consider cases with repetition.] ii. Find the number of permutations of different objects. iii. Find the number of permutations of objects from different objects. Find the number of permutations of objects comprising of identical objects of type 1, … , identical objects of type , where [Exclude: "Round table" & circular permutations] iv. v. Determine the permutation of a set of objects with conditions Find the number of ways to form combinations of objects from different objects. vi. vii. Determine the combinations of a set of objects with condition. Distinguish between situations involving permutations & combinations (a) (b) Use permutations and combinations to calculate probabilities At the end of the lesson, students should be able to A permutation is an ordered arrangement of all or part of a set of objects. E.g. Permutation of all the 3 letters has 6 outcomes as follows: {ABC, ACB, BAC, BCA, CAB, CBA } Factorial Notation where E.g. ▪ ▪ ▪ Remarks Permutations 7.2 Probability involving Permutations & Combinations C07 - SM025 Page 1

Fundamental principle of counting If there are ways to make the first selection and ways to make the second selection, then there are ways to make both the selection. E.g. A set meal is formed by a main course and a drinks based on the selection below: Main course = {Sandwich, Burgers, Hotdog} Drinks = {Coffee, Tea} There are to have a set meal. Example 1 How many arrangements of the letters A, B, and C are there? Solution: Number of arrangement Example 2 How many ways can we form a number code for this padlock? Solution: Number of arrangement 1. Sandwich + Coffee 2. Sandwich + Tea 3.Burger + Coffee 4.Burger + Tea 5. Hotdog + Coffee 6. Hotdog + Tea Permutation - I C07 - SM025 Page 2

Permutation of If out of objects are arranged, then there are ways to arrange them. [Remarks: ] E.g. Number of ways How many ways are there to arrange all the letters from A, B, C and D ? Example 3 [Permutation of out of differents objects] In how many different ways can the letters of word “LEARN” be arranged? Solution: Number of ways Example 4 [Permutation of out of differents objects with conditions] In how many different ways can the letters of word “LEARN” be arranged if the first letter and the last letter are vowels? Solution: Number of ways ABCD ABDC ACBD ACDB ADBC ADCB BACD BADC BCAD BCDA BDAC BDCA CABD CADB CBAD CBDA CDAB CDBA DABC DACB DBAC DBCA DCAB DCBA Permutation - II C07 - SM025 Page 3

Permutation of If out of objects are arranged, then there are ways to arrange them. [Remarks: ] E.g. Number of ways How many ways are there to arrange any 2 letters from A, B, C and D ? Example 5 [Permutation of out of differents objects] There are six competitors in the final of a 100m race. The first three competitors to complete the course will all receive medals. Calculate the number of different possible groups of medal winners. Solution: Number of groups Permutation of If out of objects are arranged with of them are identical, then there are ways to arrange them. [Remarks: ] E.g. Number of ways How many ways are there to arrange all the letters from A, B, B, B and C ? Example 6 [Permutation of out of objects with some identical objects] How many different permutations can be made using the letters in “MATHEMATICS”? Solution: Number of permutations AB AC AD BC BD CD BA CA DA CB DB DC BBBAC BBBCA ABBBC CBBBA ACBBB CABBB BBABC BBCBA BBACB BBCAB ABBCB CBBAB BABBC BCBBA ABCBB CBABB BACBB BCABB BABCB BCBAB Permutation - III C07 - SM025 Page 4

Example 7 [Conditional Permutation] A group of cadets consisting of 4 males and 3 females are to be seated in a row. How many ways can it be done if a) there is no restriction ? Solution: 4 males and 3 females Number of ways b) all the female cadets are to sit together? Solution: 4 males and 3 females Number of ways c) all the female cadets must be separated? Solution: 4 males and 3 females Number of ways d) the male and female cadets sit at alternate places? Solution: 4 males and 3 females Number of ways Permutation - IV C07 - SM025 Page 5

Example 8 [Conditional Permutation] Without repetition, how many 3-digit number can be formed by using the digits 2, 3, 5, 6, 7, 9 if the number is a) less than 400 Solution: 2, 3, 5, 6, 7, 9 Number of 3-digit number b) an odd number Solution: 2, 3, 5, 6, 7, 9 Number of 3-digit number c) less than 400 and an odd number Solution: 2, 3, 5, 6, 7, 9 Number of 3-digit number Permutation - V C07 - SM025 Page 6

Example 1 In how many ways could a quiz team of four be chosen from a group of fifteen students? Solution: 15 students Number of ways Example 2 A team of 7 players is to be chosen from a group of 12 players. One of the seven is then to be elected captain and another one is to be the vice-captain. In how many ways can this be done? Solution: 12 players Number of ways Example 3 Five students were chosen from a group of eight boys and five girls. In how many ways could the group be chosen if there are to be more boys than girls in that group? Solution: 8 boys & 5 girls Number of ways Combination - I Tuesday, January 11, 2022 9:29 PM C07 - SM025 Page 7

Example 4 A committee consisting of three members is to be formed from a group of 25 including five women. How many different committees can be chosen: (a) without any restrictions Solution: 25 people (5 women & 20 men) Number of committees (b) if there must be at least a woman. Solution: 25 people (5 women & 20 men) Number of committees Example 5 (a) Find the number of ways to arrange 3 letters from the word JANUARY? Given the word JANUARY . (b) Find the number of ways to select 3 letters from the word JANUARY? Solution: JANUARY Number of ways Solution: JANUARY Number of ways Combination - II Tuesday, January 11, 2022 9:29 PM C07 - SM025 Page 8

Example 6 (a) without any restrictions In a mathematics examination, students will be given 8 questions. Students have to answer only 5 questions. Find how many ways the student may choose their questions Solution: 8 Questions Choose 5 Questions Number of ways (b) if question 2 and 4 are compulsory Solution: 8 Questions Choose 5 Questions Number of ways (c) if question 5 is chosen, then question 6 should be ignored. Solution: 8 Questions Choose 5 Questions Number of ways Combination - III Tuesday, January 11, 2022 9:29 PM C07 - SM025 Page 9

A four-digit PIN is selected. What is the probability that there are no repeated digit. Solution: Let the event of there are no repeated digit arranged together. Example 1 C07 - SM025 Page 10

Three red marbles, four yellow marbles and two green marbles are arranged in one row on a table. Find the probability (a) all the four yellow marbles must be arranged together. (b) all the four yellow marbles not together. (c) the green marbles must be in the first and last position of the row. Solutions: (a) Let the event of all the four yellow marbles must be arranged together. (b) Let the event of all the four yellow marbles not together (c) Let the event of the green marbles must be in the first and last position of the row. Example 2 C07 - SM025 Page 11

4 letters are chosen randomly from the word COMPUTER . Find the probability of (a) all the four letters chosen are consonant (b) the letter C must be chosen (c) the letters M and P must be chosen simultaneously Solutions: (a) Let the event of all the four letters chosen are consonant (b) Let the event of the letter must be chosen (c) Let the event of the letters and must be chosen Example 3 C07 - SM025 Page 12

(a) men, women and children are selected one each (b) at least one child is selected. At a local clinic, there are 8 men, 5 women and 3 children in the waiting room. If three patients are randomly selected, find the probability Solutions: (a) Let the event of 1 man, 1 woman and 1 child are selected (b) Let the event of at least one child is selected Example 4 C07 - SM025 Page 13