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2 Warm Up 1.List all combinations of Roshanda, Shelli, Toshi, and Hector, taken three at a time 2. In how many ways can 5 people be seated in a row of 5 chairs?

Copyright © Ed2Net Learning Inc.3 4. In how many ways can 3 representatives be chosen from a group of 11 people? 5. In how many ways can a 5player team be chosen from 16 people? 3. In how many ways can three clarinet players be seated in first, second, and third seats in the orchestra? Warm Up

Copyright © Ed2Net Learning Inc.4 Let’s summarize what we have learnt in the previous lesson 1. The fundamental counting principle says that, the the total number of different ways a task can occur is: n1 * n2 * n3 * ………………… * nr where n1, n2, n3..nr are the tasks performed at first, second and rth stages 2. Factorial is a shorthand notation for a multiplication process. The Fundamental Counting Principle says that the number of ways to choose n items is n ! n! = n * (n-1) * (n-2) *……………*3 * 2 * 1 3. Any factorial less than n! is a factor of n!

Copyright © Ed2Net Learning Inc.5 5. The number of ways (or combinations) in which r objects can be selected from a set of n objects, where repetition is not allowed, is denoted by, or C(n, r) n C r n! n C r = (n-r)! r! 6. Pascal’s Triangle illustrates the symmetric nature of a combination. i,e C(n,r) = C(n,n-r) Recap 4. A combination focuses on the selection of objects without regard to the order in which they are selected

Copyright © Ed2Net Learning Inc The sum of all combinations : C(n,0) + C(n, 1) C(n, n) = 2^n 11. Relationship between permutation and combination is p n r n C r = r! Recap 7. C(n, 1) = n 8. C(n, 0) = 1 9. C(n, n) = 1

Copyright © Ed2Net Learning Inc.7 Concept of Permutation: The letters a, b and c can be arranged in six different ways abc acb bac bca cab cba Each of these arrangements is called the permutations of the letters a, b and c

Copyright © Ed2Net Learning Inc.8 The symbol n P r is used to indicate the number of permutations n objects taken r at a time. n P r = n (n -1)(n -2)…….[n-(r-1)]=n!/ (n - r)! When r = n, last factor of the in the above equation n-(r-1) = n-(n-1) = 1 n P r = n (n -1)(n -2)…….1= n! r factors

Copyright © Ed2Net Learning Inc.9 A permutation focuses on the arrangement of objects with regard to the order in which they are arranged. A combination focuses on the selection of objects without regard to the order in which they are selected. The distinction between a combination and a permutation has to do with the sequence or order in which objects appear. Permutation and Combination - Difference

Copyright © Ed2Net Learning Inc.10 Since the order does not matter in combinations, there are clearly fewer combinations than permutations. The combinations are contained among the permutations. They are a subset of the permutations Relation between permutation and combination

Copyright © Ed2Net Learning Inc.11 Example Find n P r for n = 7 and r = 5. n P r = 7 P 5 = = = /2.1 = = 720 nPr = n!/(n-r)! 7! (7 – 5)! 7! 2! n! (n – r)!

Copyright © Ed2Net Learning Inc.12 Example Find n P r for n = 4 and r = 4. n P n = n! 4 P 4 = 4! = = 24 nPn = n!

Copyright © Ed2Net Learning Inc.13 Example Find the number of permutations of the three letters b, c and d. Solution: The number of permutations of n objects is n! n P n = n! 3 P 3 = 3! = = 12 The number of Permutations of n objects is n!

Copyright © Ed2Net Learning Inc.14 Example From a set of 8 different books, 5 are to selected and arranged on a self. How many arrangements are possible? Solution: Find the number of permutations of 8 books taken 3 at a time: n P r = 8 P 5 = 8!/(8-5)! = /3.2.1 = = 6720 The number of permutations of n objects taken r at a time is nPr. n! (n – r)!

Copyright © Ed2Net Learning Inc.15 If a set of n elements has n 1 elements of one kind, n 2 of another kind alike and so on, then the number of permutations, P, of the n elements taken n at a time is given by: P = n! n 1 !.n 2 !

Copyright © Ed2Net Learning Inc.16 Example Find the number of ways the letters in the word BEEKEEPER can be arranged? Solution: There are 9 letters, 5 of which are E’s, 1 is B, 1 is K and 1 is P. P = P = 9!/5!.1!.1! = / = = 3024 P = n! n 1 !.n 2 ! n! n 1 !.n 2 !...

Copyright © Ed2Net Learning Inc.17 Your Turn Evaluate: 1.5! 2.10!/8! 3.6!/(6-2)! 4. 5 P P 3

Copyright © Ed2Net Learning Inc.18 Your Turn Tell the number of ways the letters in each word can be arranged. 6.TOP 7.NOON 8.SUPPER 9.In how many ways 7 people can be lined up in a row? 10. In how many ways 6 different books can be arranged on a self?

Copyright © Ed2Net Learning Inc.19 Refreshment Time

Copyright © Ed2Net Learning Inc.20 Lets play a Game

Copyright © Ed2Net Learning Inc In how many ways can 3 cards from a deck of cards be laid in a row face up?

Copyright © Ed2Net Learning Inc a) How many vowels are there? How many consonants are there? b) In how many ways can you choose at random a vowel and a consonant?

Copyright © Ed2Net Learning Inc Show that 6 P 4 = 6( 5 P 3 )

Copyright © Ed2Net Learning Inc.24 Let Us Review The symbol n P r is used to indicate the number of permutations n objects taken r at a time. n P r = n!/ (n - r)! The numbers of n objects is n!. If a set of n elements has n 1 elements of one kind, n 2 of another kind alike and so on, then the number of permutations, P, of the n elements taken n at a time is given by: P = n! n 1 !.n 2 !

Copyright © Ed2Net Learning Inc.25 You did Great !!!! Keep up the good work