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# Quantitative techniques basics of mathematics permutations and combinations_part ii_30 pages

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• 1. Basic Quantitative Techniques ABS-Bangalore Basic Quantitative Techniques - RVMReddy - ABS July 14, 2010
• 2.
• Dr. R. Venkatamuni Reddy Associate Professor
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