Concept of Permutations

Theory:

• Factorial Notation ! Or L
n! = n(n - 1)(n - 2)(n - 3) . . . . . 3.2.1 = product of n consecutive integers starting from 1 to the number n.
• 0! = 1.
• Factorials of only Natural numbers are defined.
n! is defined only for n ≥ 0.
n! is not defined for n < 0.

Permutations:

Each of the different arrangements which can be made by taking some or all of a number of given things or objects at a time is called a permutation.

Permutations (represented by ⁿP_r):

Permutation means arrangement of things. It is defined as the number of ways in which r things at a time can be selected and arranged amongst n things.
ⁿP_r = number of permutations (arrangement) of n things taken r at a time. Thus the symbol ⁵P₂ will denote the number of permutation or arrangement of 5 different things taken 2 at a time.
$$\eqalign{ & ^{\text{n}}{{\text{P}}_{\text{r}}} = \frac{{{\text{n}}!}}{{\left( {{\text{n}} - {\text{r}}} \right)!}};\,\,{\text{n}} \geqslant {\text{r}} \cr & ^{\text{5}}{{\text{P}}_{\text{2}}} = \frac{{5!}}{{3!}} \cr} $$

Situation in which permutations are used:
• Making words and numbers from a set of available letters and digits respectively.
• Filling post with people.
• Selection of batting order of a cricket team of 11 from 16.
• Putting distinct objects/people in distinct places. For example, making people sit, putting letters in envelopes, finishing order in horse race etc.

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