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    Quant05 Quant05 Presentation Transcript

    • Chapter 5. Basic Concepts of Permutations and Combinations
      The factorial n, written as n!, represents the product of all integers from 1 to n both inclusive.
      Hence, n! = n (n – 1) (n – 2) (n – 3)…. 3.2.1
      0! = 1
       
      Permutation
      The ways of arranging or selecting smaller or equal number of persons or objects from a group of persons or collections of objects with due regard being paid to the order of arrangement or selection, are called permutations.
      Permutations of n things chosen r at a time, nPr = n (n – 1) (n – 2)… (n – r + 1)
      Revision Notes – Quantitative Aptitude
      www.cptsuccess.com
      Page 1 of 1
    • Chapter 5. Basic Concepts of Permutations and Combinations
      Properties of Permutation
      Permutations of n things taken all n things at a time, nPn= n!
      nPr in factorial notation is written as n! / (n – r)!
       
      Circular Permutation
      The number of circular permutations of n different things chosen at a time is (n – 1)!
      The number of ways of arranging n persons along a round table so that no person has the same two neighbors = ½ (n – 1)!
      The number of necklaces formed with n beads of different colors = = ½ (n – 1)!
       
      Revision Notes – Quantitative Aptitude
      www.cptsuccess.com
      Page 1 of 1
    • Chapter 5. Basic Concepts of Permutations and Combinations
      Permutations with different conditions applied
      Number of permutations of n distinct objects when a particular object is not taken in any arrangement is n-1Pr
      Number of permutations of n distinct objects when a particular object is always included in the arrangement is r. n-1Pr – 1
       
      Combinations
      The number of ways in which smaller or equal number of things are arranged or selected from a collection of things where the order of selection or arrangement is not important are called combinations
      Number of combinations of n different things taken r at a time = nCr given that 0 ≤ r ≤ n
      nCr = n! / r! (n – r)!
      Revision Notes – Quantitative Aptitude
      www.cptsuccess.com
      Page 1 of 1
    • Chapter 5. Basic Concepts of Permutations and Combinations
      Properties of nCr
      nCr = nCn – r (given that 0 ≤ n – r ≤ n)
      n+1Cr = nCr + nCr-1
      nC0 = 1
      nCn = 1
       
      Permutation when some things are alike, all taken at a time
      P = n! / (p! q! r!) where n is the number of things, p number of the things are exactly of one kind, q of exactly another kind, r of a third kind and the remaining things different.
      Permutation when each things may be repeated upto r times in an arrangement = nr 
      Revision Notes – Quantitative Aptitude
      www.cptsuccess.com
      Page 1 of 1
    • Chapter 5. Basic Concepts of Permutations and Combinations
      Combination of n different things taken some or all of n things at a time = 2n – 1
      Combination of n things taken some or all at a time when p of the things are alike of one kind, q of the things are alike and of another kind and r of the things are alike of a third kind = [(p + 1) (q + 1)(r + 1)….] – 1
      Combination of selecting r1 things from a set of n1 objects and r2 things from a set of n2 objects where combination of r1 things and r2 things are independent is given by n1Cr1 x n2Cr2
       
      Revision Notes – Quantitative Aptitude
      www.cptsuccess.com
      Page 1 of 1