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


The different arrangements of a given number of things by taking some or all at a time, are called permutations.
  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:

  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


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.
  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.

Number of Combinations

The number of all combinations of n things, taken r at a time is:



  1. nCn = 1 and nC0 = 1.
  2. nCr = nC(n - r)
  1. 16C13 = 16C(16 - 13) = 16C3

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