6 1 1 Combinations Example 1 List out

6. 1. 1 & Combinations

Example 1 List out the different three-digit numbers can be made out of the digits 1, 2, 3, if no digit is repeated. 123 132 213 231 312 321 6 different numbers 123 1 selection Permutation The number of arrangement possible for a set of objects s. An event where ORDER does matter Combination The number of selection of objects s An event where ORDER does NOT matter

Permutations of n different Example 2 objects In how many different orders can you place four textbooks on your shelf ? Number of ways = 4 x 3 x 2 x 1 =4! = 24 4 3 2 1 1 st 2 nd 3 rd 4 th Number of arrangements of n different objects in a row is

Permutations of n different objects, taking r objects each time ( r n ) Example 3 How many 4 -digit numbers can you form with the digits 1, 2, 3, 4, 5, 6, if all the digits used in a number are to be different? 6 5 4 3 Number of ways = 6 x 5 x 4 x 3 1 st 2 nd 3 rd 4 th = 360 Number of arrangements of n different objects taking r objects each time in a row is

Permutations when there are similar Example 4 objects List out the different three-digit numbers can be made out of the digits 1, 3 113 113 131 131 Number of ways 311 311 But we have 2 similar numbers Number of ways =3 =3! =6 1, 1 Number of similar objects In general, if there a total of n objects that consist of p same objects, q another same objects, r another same objects and so on, the number of arrangements in a row is

Example 5 How many password consists of 6 letters are possible by using the letters AAABBC ? Number of passwords = 60 Number of similar objects Example 6 In how many ways can arrange 3 out of the 6 cards in a row from the cards 1, 2, 2, 3, 4 and 5 ? Case 1 : Only 1 of the card with digit 2 is taken Number of ways = 60 Case 2 : Both the cards with digit 2 are taken Number of ways = 12 Total number of ways = 60 + 12 = 72 Number of similar objects

Permutations in a Example 7 circle In how many ways can you plant five different species of flowers around a circular patio? A E B D E E A A C ……. B D C C D These 2 arrangements are technically the same Number of ways = 4 ! = ( 5 – 1 ) ! Number of arrangements of n different objects in a circle is

Permutation with Example 8 Conditions Given the digits 0, 2, 3, 5, 8 and 9, you are required to form 4 -digit numbers that cannot start with 0. Find the number of (a) even numbers that can be formed, 156 (b) numbers bigger than 5000, 180 (c) numbers which are even and bigger than 5000. 96 Example 9 Four male and two female film stars are arranged in different ways in a straight line for several photographs to be taken. How many different arrangements are possible if (a) there are no restrictions, 720 (b) two particular male film stars must stand together, 240 (c) the two female film stars must be separated? 480 Exercise 6. 2 pg. 251 Question 1 – 7

Combinations

Combination The number of selection of objects s An event where ORDER does NOT matter The number of combinations ( selections ) of the n objects taking r objects at a time is Exercise 6. 2 pg. 251 Question 8 – 12 Example 10 Club AAA is to have 9 committee members chosen among 3 teachers, 8 boys and 6 girls. In how many ways can this committee be formed if it has to have (a) 1 teacher in the committee, (b) not more than 2 teachers in the committee, (c) no teacher but more boys than girls in the committee. Example 11 ( CD Teaching Courseware )