PERMUTATIONS AND COMBINATIONS

Test
Materials
Please Read the below formula and concept every time before you start to solve the questions.
  • 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.
     
  • 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 abc by taking two at a time are (abbaaccabccb).
    2. All permutations made with the letters abc taking all at a time are: ( abcacbbacbcacabcba)
  • 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. Cor. number of all permutations of n things, taken all at a time = n!.
  • 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!)
    Combinations:
  • Each of the different groups or selections 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 abc taking abbcca.
    3. The only combination that can be formed of three letters abc 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.
  • Number of Combinations:The number of all combinations of n things, taken r at a time is:
    nCr = n! = n(n – 1)(n – 2) … to r factors .
    (r!)(n – r)! r!
      Note:
    1. nCn = 1 and nC0 = 1.
    2. nCr = nC(n – r)

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