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Quantitative Aptitude :: Permutations Formulas

Permutations Questions and Answers

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  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.

  1. 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)

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.
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:
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 a, b, c taking ab, bc, ca.
The only combination that can be formed of three letters a, b, c 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:
i. 11C4 = (11 x 10 x 9 x 8)/(4 x 3 x 2 x 1) = 330.
ii. 16C13 = 16C(16 - 13) = 16C3 = (16 x 15 x 14)/3! = (16 x 15 x 14)/(3 x 2 x 1) = 560.