Permutation and Combination - Quantitative Aptitude - Aptitude question and answers

Explanation / Important formulas:

Permutation implies arrangement where order of things is important and includes word formation number formation, circular permutation etc.

  • All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).
  • All permutations made with the letters a, b, c taking all at a time are (abc, acb, bac, bca, cab, cba).

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:

  • 6P2 = (6 x 5) = 30.
  • 7P3 = (7 x 6 x 5) = 210.

More information: https://en.wikipedia.org/wiki/Permutation

Combination means selection where order is not important and it involves selection of team forming geometrical figures, distribution of things etc.

Examples:

  • 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]
  • All the combinations formed by abc taking ab, bc, ca.
  • The only combination that can be formed of three letters abc taken all at a time is abc.
  • Various groups of 2 out of four persons A, B, C, D are: AB, AC, AD, BC, BD, CD.
  • 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 ! / [(r!)(n-r)]! = [n(n-1)(n-2)…. To r factors] / r!

Note:  nCn = 1 and nC0 = 1 , nCr = nC(n – r) 

Examples:

  • 11C4 = (11 x 10 x 9 x 8) / (4 x 3 x 2 x 1) = 330
  • 16C13 = 16C(16 – 13) = 16C3 = 16 x 15 x 14 / 3!= (16 x 15 x 14) / (3 x 2 x 1) = 560

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:                                 

  • We define 0! = 1.
  • 4! = (4 x 3 x 2 x 1) = 24.
  • 5! = (5 x 4 x 3 x 2 x 1) = 120.

More information: https://en.wikipedia.org/wiki/Combination

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