Quantitative techniques basics of mathematics permutations and combinations_part ii_30 pages
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Quantitative techniques basics of mathematics permutations and combinations_part ii_30 pages Quantitative techniques basics of mathematics permutations and combinations_part ii_30 pages Presentation Transcript

  • Basic Quantitative Techniques ABS-Bangalore Basic Quantitative Techniques - RVMReddy - ABS July 14, 2010
    • Dr. R. Venkatamuni Reddy Associate Professor
    • Contact: 09632326277, 080-30938181 [email_address] [email_address]
    Basic Quantitative Techniques - RVMReddy - ABS July 14, 2010
  • Permutations and Combinations July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  • 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
    • “ 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
    • 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
    • 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
    • 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
    • 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
  • Permutations Basic Quantitative Techniques - RVMReddy - ABS
  • 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
    • 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
    • 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
    • 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
    • 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
  • Combinations Basic Quantitative Techniques - RVMReddy - ABS
  • Combinations Basic Quantitative Techniques - RVMReddy - ABS
  • 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
  • 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
  • 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
  • Combination formula proof
    • Note that the textbook explains it slightly differently, but it is same proof
    July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
    • 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
  • 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
  • Binomial Coefficients Basic Quantitative Techniques - RVMReddy - ABS
  • Binomial Coefficients Basic Quantitative Techniques - RVMReddy - ABS
  • 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
  • Binomial Coefficients July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
  • Basic Quantitative Techniques - RVMReddy - ABS
  • 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
  • Thank You Basic Quantitative Techniques - RVMReddy - ABS July 14, 2010