This document discusses quantitative techniques including permutations, combinations, and binomial coefficients. It provides formulas and examples for calculating permutations and combinations when arranging objects in different orders or selecting subsets of objects irrespective of order. It also discusses Pascal's triangle and using the divide-and-conquer technique to efficiently calculate binomial coefficients.

1. Basic Quantitative Techniques ABS-Bangalore Basic Quantitative Techniques - RVMReddy - ABS July 14, 2010

2. Dr. R. Venkatamuni Reddy Associate Professor Contact: 09632326277, 080-30938181 [email_address] [email_address] Basic Quantitative Techniques - RVMReddy - ABS July 14, 2010

3. Permutations and Combinations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

4. Permutations Permutations refers to the different ways in which a number of a number of objects can be arranged in a different order Example: Suppose there are two things x and y, they can be arranged in to two different ways i.e,. xy and yx . These two arrangements is called permutation Similarly x, y and z xyz, xzy, yxz, yzx, zxy, zyx is 6 arrange permutation (if we want to have two things only from x,y,z then xy,xz,yz,yx,zx,yz only in this case) July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

5. “ The word permutation thus refers to the arrangements which can be made by taking some or all of a number of things” Formulae 1: Finding the number of permutations of ‘n’ dissimilar things taken ‘r’ at a time n=number of different things given, r=number of different things taken at a time out of different things given Permutations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

6. Example 1: There are six boxes and three balls. In how many ways can these three balls be discretely put into these six boxes. Solution: Example 2: How many four-letter words can be made using the letters of the word ‘BANGALORE’ and ‘ALLIANCE’ Solution: n=9, r=4 and n=8, r=4 Permutations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

7. Example 3: How many arrangements are possible of the letters of the words ‘JAIPUR’, ‘BANGALORE’ and ‘ALLIANCE’ Hint: n=6, r=6 and n=9, r=9 and n=8,r=8 Permutations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

8. Formulae 2: Finding the number of permutations of ‘n’ things taken ‘r’ at a time, given that each of the elements can be repeated once, twice….up to ‘r’ times Or ‘ n’ things taken all at a time of which ‘p’ are alike, ‘q’ others are alike and ‘r’ others alike Example 1: How many permutations are possible of the letters of the word PROBABILITY when taken all at a time? Permutations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

9. Solution: n=11, p=2 ( as letter B is occurring twice in the given word) , and q=2 ( as letter I is occurring twice in the given word) And all other letters in the given word are different. The required number of permutations is (r is not valid in this) Example 2: You are given a word “MANAGEMENT” and asked to compute the number of permutations that you can form taking all the letters from this word? Permutations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

10. Permutations Basic Quantitative Techniques - RVMReddy - ABS

11. Permutation formula proof There are n ways to choose the first element n -1 ways to choose the second n -2 ways to choose the third … n - r +1 ways to choose the r th element By the product rule, that gives us: P ( n , r ) = n ( n -1)( n -2)…( n - r +1) July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

12. Combinations refers to the number of arrangements which can be made from a group of things irrespective of the order Combinations differ from permutations in that one combination such as xyz may be stated in the form of several permutations just by rearranging the orders as : xyz, xzy, yxz, yzx, zxy, zyx Note: All of these are one combination but they are six permutations IMP Note: The number of permutations is always greater than the number of combinations in any given situation since a combination of n different things can be generate n factorial permutations Combinations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

13. Formulae 1: The number of r -combinations of a set with n elements, where n is non-negative and 0≤ r ≤ n is: n= number of different things given r= number of different things taken at a time out of different things given Combinations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

14. Example 1: in how many ways can four persons be chosen out of seven? n=7, r=4 Example 2: Find the number of combinations of 50 things taking 46 at a time. ANS: 230300 Combinations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

15. Formulae 2: The number of ways in which x+y+z things can be divided into three groups contain x, y, and z things respectively is Example: In how many ways can 10 books be put to three shelves which can contain 2, 3 and 5 books respectively? Combinations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

16. Combinations Basic Quantitative Techniques - RVMReddy - ABS

17. Combinations Basic Quantitative Techniques - RVMReddy - ABS

18. Combinations How many different poker hands are there (5 cards)? How many different (initial) blackjack hands are there? July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

19. Combination formula proof Let C (52,5) be the number of ways to generate unordered poker hands The number of ordered poker hands is P (52,5) = 311,875,200 The number of ways to order a single poker hand is P (5,5) = 5! = 120 The total number of unordered poker hands is the total number of ordered hands divided by the number of ways to order each hand Thus, C (52,5) = P (52,5)/ P (5,5) July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

20. Combination formula proof Let C ( n , r ) be the number of ways to generate unordered combinations The number of ordered combinations (i.e. r -permutations) is P ( n , r ) The number of ways to order a single one of those r -permutations P ( r,r ) The total number of unordered combinations is the total number of ordered combinations (i.e. r -permutations) divided by the number of ways to order each combination Thus, C ( n,r ) = P ( n,r )/ P ( r,r ) July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

21. Combination formula proof Note that the textbook explains it slightly differently, but it is same proof July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

22. Let n and r be non-negative integers with r ≤ n . Then C ( n , r ) = C ( n , n-r ) Proof: Combination formula proof July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

23. Binomial Coefficients The expression x + y is a binomial expression as it is the sum of two terms. The expression (x + y) n is called a binomial expression of order n . Basic Quantitative Techniques - RVMReddy - ABS

24. Binomial Coefficients Basic Quantitative Techniques - RVMReddy - ABS

25. Binomial Coefficients Basic Quantitative Techniques - RVMReddy - ABS

26. Binomial Coefficients Pascal’s Triangle The number C(n , r) can be obtained by constructing a triangular array. The row 0, i.e., the first row of the triangle, contains the single entry 1 . The row 1, i.e., the second row, contains a pair of entries each equal to 1 . Calculate the n t h row of the triangle from the preceding row by the following rules: Basic Quantitative Techniques - RVMReddy - ABS

27. Binomial Coefficients July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

28. Basic Quantitative Techniques - RVMReddy - ABS

29. Binomial Coefficients The technique known as divide and conquer can be used to compute C(n , r ). In the divide-and-conquer technique, a problem is divided into a fixed number, say k , of smaller problems of the same kind. Typically, k = 2 . Each of the smaller problems is then divided into k smaller problems of the same kind, and so on, until the smaller problem is reduced to a case in which the solution is easily obtained. The solutions of the smaller problems are then put together to obtain the solution of the original problem. July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS

30. Thank You Basic Quantitative Techniques - RVMReddy - ABS July 14, 2010