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# 3. permutation and combination

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### 3. permutation and combination

1. 1. PERMUTATION & COMBINATION
2. 2. MULTIPLICATION RULE <ul><li>If an operation can be performed in r different ways and a second operation can be performed in s different ways, then both the operations can be performed successively in r x s ways. </li></ul>
3. 3. Multiplication Rule/Principle Adidas Reebox Nike 10 23 7 9 2 3 4   24 = The number of ways the outfit can be chosen =
4. 4. PERMUTATION <ul><li>In permutation, the order of the objects or outcomes is important. Each different order represents a different outcome. </li></ul>Permutation – variation, order Arrangement
5. 5. PERMUTATION Permutation – the arrangement is important <ul><li>How many ways can the letter </li></ul><ul><li>X and Y be arranged? </li></ul>X Y X Y There are two ways  two different permutation
6. 6. COMBINATIONS <ul><li>In Combinations, we do not arrange the selections in order. </li></ul>Combination – grouping,selection Choices
7. 7. Combination <ul><li>Arrangement is not important </li></ul>X Y Or X Y Are the same  one combination
8. 8. Tom & Jerry Jerry & Tom How many arrangements? 2 PERMUTATIONS
9. 9. Tom & Jerry Jerry & Tom They are the same cat & the mouse OR How many combinations? 1 COMBINATION
10. 10. DIFFERENCES BETWEEN PERMUTATIONS AND COMBINATIONS PERMUTATIONS COMBINATIONS Arranging people, digits, numbers, alphabets, letters, colours. Keywords: Arrangements, arrange,… Selection of menu, food, clothes, subjects, teams. Keywords: Select, choice,…
11. 11. Permutation <ul><li>Number of ways to arrange </li></ul><ul><li>n different objects </li></ul>
12. 12. Number of ways to arrange 3 different objects A B C A B C B B A C C A C A B C B A 6 ways A B C
13. 13. Number of ways to arrange 4 different objects A B C D B C D 24 A B C D A B C D A B C D A B C D A B C D A B C D
14. 14. Number Number of of objects ways 1 ……… ……………………… 2 ………. …….………………… 3 ……….. ……………….……… 4 ……….. ……………….……… 1 2 x 1 1 2 6 3 x 2 2 x 1 24 4 x 6 3 x 2 x 1 120 5 x 24 4 x 3 x 2 x 1 6 x 5 x 4 x 3 x 2 x 1 = 6! = 5! = 4! = 3! = 2! = 1! The number of ways to arrange n objects = n ! 5 ……….. ………………………. 6 ……….. ………………………. Factorial
15. 15. 1. To arrange 10 different objects = 10 ! 2. To arrange digits 2, 5, 6, 8 = 4 ! 4 different objects 3. To arrange 12 finalists 12 different objects = 12 !
16. 16. DNA 23 pairs of chromosomes 23 ! different ways to arrange
17. 17. The number of ways to arrange 23 different objects ? 25852016738884976640000 23 ! = 2.6 x 10 22
18. 18. Permutation <ul><li>Number of ways to arrange </li></ul><ul><li>r objects from n objects </li></ul>
19. 19. 8 choices 7 choices 6 choices 5 choices 4 choices Number of ways to arrange 5 students from 8 students. 8 x 7 x 6 x 5 x 4 = 6720
20. 20. 8 choices 7 choices 6 choices 5 choices 4 choices number of ways to arrange r from n objects n (n-1) (n-2) (n-3) (n-(r-1))
21. 21. 1. Questions related to Forming Numbers with digits and conditions <ul><li>Use Multiplication Rules </li></ul>
22. 22. Condition 1 Find the number of ways to form 5 letter word from the letters W, O,R, L, D, C, U, P with the condition that it must starts with a vowels. is filled first W R L D C P O O U 2 7 6 5 4 U =1680
23. 23. Find the number of ways to form 6 letter word from the letters B, E, C, K, H, A, M with the condition that it must starts with a consonant. B CKHM 5 EA 6 5 5x6x5x4x3x2 = 1200 4 3 2
24. 24. 2. Questions related to Forming Numbers with digits and conditions <ul><li>Use Multiplication Rules </li></ul><ul><li>Conditions : </li></ul><ul><li>Sit side by side , next to each other – group together and consider as 1 object for arranging with other objects, make sure remember the arrangement of the grouped objects itself. </li></ul>
25. 25. A E A E P N L T Y P N L T Y A E 6 ! 2! = 1440 4. To arrange PENALTY such that vowels are side by side 1 2 3 4 5 6 
26. 26. 3. Complimentary Methods <ul><li>Use: </li></ul><ul><li>The number of arrangements of event A </li></ul><ul><li>= Total arrangements – arrangement of A’ </li></ul>A A’ S
27. 27. Example <ul><li>Find the number of the arrangement of all nine letters of word SELECTION in which the two letters E are not next to each other. </li></ul><ul><li>Solutions: </li></ul><ul><li>Total no. of arrangements – No. of arrangements with two E next to each other </li></ul>
28. 28. Combinations n objects choose n = 1
29. 29. N = 4 Choose 1: A B C D A B C D Choose 2: A B A C A D B C Choose 3: A B C A C D B C D B D C D A B D Choose 4: A B C D = 4 = 4 C 1 = 6 = 4 C 2 = 4 = 4 C 3 = 1 = 4 C 4
30. 30. Combinations
31. 32. Conditional Combination 1 A football team has 17 local players and 3 imported players. Eleven main players are to be chosen with the condition that it must consist of 2 imported players. Find the number of ways the main player can be chosen. import local 3 17 2 9 3 C 2 17 C 9   = 72930 r n
32. 33. Condition Combination 2 A committee consisting of 6 members is to be chosen from 3 men and 4 women. Find the number of ways at least 3 women are chosen. W3 M3 , or W4 M2 , 4 C 3 X 3 C 3 + 4 C 4 X 3 C 2 = 7
33. 34. CONCLUSIONS <ul><li>1. Compare and Contrast between Permutations and Combinations. </li></ul>
34. 35. DIFFERENCES BETWEEN PERMUTATIONS AND COMBINATIONS PERMUTATIONS COMBINATIONS 1. Order is importent 2. Arranging people, digits, numbers, alphabets, letters, colours, … 3. Keywords: Arrangements, arrange,… <ul><li>Order is not important. </li></ul><ul><li>2. Selection of menu, food, clothes, subjects, teams, … </li></ul><ul><li>Keywords: </li></ul><ul><li>Select, choice,… </li></ul>
35. 36. 2. Formula <ul><li>Difference between the two formulae: </li></ul><ul><li>Use the calculator to find the values of permutations and combinations. </li></ul>
36. 37. 3. If not sure, try to use the Multiplication Rules <ul><li>Know the ways how to handle conditions like: </li></ul><ul><li>Sit side by side , next to each other, even/odd numbers, more/less than, starts/ends with vowel/consonants, … </li></ul>
37. 38. Find the number of ways to form 6 letter word from the letters B, E, C, K, H, A, M with the condition that it must starts with a consonant. B CKHM 5 EA 6 5 5x6x5x4x3x2 = 1200 4 3 2
38. 39. 4. For complicated cases: <ul><li>Simplify by using Complimentary Methods </li></ul><ul><li>The number of arrangements of event A </li></ul><ul><li>= Total arrangements – arrangement of A’ </li></ul>A A’ S
39. 40. PROBABILITY

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