Home » Aptitude » Permutation and Combination
  1. Factorial Notation: Let n be a positive integer. Then, factorial n, denoted n! is defined as:

    n! = n(n - 1)(n - 2) ... 3.2.1. Examples:

    1. We define 0! = 1.
    2. 4! = (4 x 3 x 2 x 1) = 24.
    3. 5! = (5 x 4 x 3 x 2 x 1) = 120.
  2. Permutations: The different arrangements of a given number of things by taking some or all at a time, are called permutations. Examples:
    1. All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).
    2. All permutations made with the letters a, b, c taking all at a time are:
      ( abc, acb, bac, bca, cab, cba)
  3. Number of Permutations: Number of all permutations of n things, taken r at a time, is given by:

     

    nPr = n(n - 1)(n - 2) ... (n - r + 1) =n!
    (n - r)!
    Examples:
    1. 6P2 = (6 x 5) = 30.
    2. 7P3 = (7 x 6 x 5) = 210.
    3. number of all permutations of n things, taken all at a time = n!.
  4. An Important Result: If there are n subjects of which p1 are alike of one kind; p2 are alike of another kind; p3 are alike of third kind and so on and pr are alike of rth kind,
    such that (p1 + p2 + ... pr) = n.
    Then, number of permutations of these n objects is =n!
    (p1!).(p2)!.....(pr!)
  5. Combinations: Each of the different groups or sections which can be formed by taking some or all of a number of objects is called a combination. Examples:
    1. Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA. Note: AB and BA represent the same selection.
    2. All the combinations formed by a, b, c taking ab, bc, ca.
    3. The only combination that can be formed of three letters a, b, c taken all at a time is abc.
    4. Various groups of 2 out of four persons A, B, C, D are:

      AB, AC, AD, BC, BD, CD.

    5. Note that ab ba are two different permutations but they represent the same combination.
  6. Number of Combinations: The number of all combinations of n things, taken r at a time is:

     

    nCr =n!=n(n - 1)(n - 2) ... (n - r + 1)  .
    (r!)(n - r)!r!
    Note:
    1. nCn = 1 and nC0 = 1.
    2. nCr = nC(n - r)
    Examples:

     

    i.   11C4 =(11 x 10 x 9 x 8)= 330.
    (4 x 3 x 2 x 1)

     

    ii.   16C13 = 16C(16 - 13) = 16C3 =16 x 15 x 14=16 x 15 x 14= 560.
    3!3 x 2 x 1