In probability theory, the arrangement of items into different ways is very important. There are two methods, permutation and combination.

Mathematically if the order (arrangement) does not matter, it is called combination and if the order (arrangement) does matter it is called permutation.

A permutation (act of permuting) is an ordered arrangement (rearrangement) of a set of items or objects. Or we can define it as

Possible ways in which a finite number of items can be arranged in different sequences with the consideration of the order of their occurrence are called permutation. Consider the three letters a, b and c. one possible permutation of these letters is abc, another permutation is acb. All the different permutations of these letters are:

Abc acb bca bac cab cba

These three objects (letters) are arranged in six possible different way (3!=6).

Rule 1:

The number of permutations of *n* different items taken *n* at a time can be denoted by nPn, where

nPn = n(n – 1) (n – 2) . . . 2.1

The formula shows that in selecting the first of the n items, there are n choices. Once the first item is selected, there remains n – 1 choices for the second item or n (n – 1) possible choices for the first two items. For third item there are n – 2 possible choice. It continues until all the item are selected in this fashion. We can write nPn in factorial notation as

nPn = n!

Rule 2:

The number of permutations of n different items taken r at a time is denoted by nPr, where

\[^n P_r= n(n-1)(n-2)\cdots (n-r+1)\]

Or

\[^nP_r=\frac{n!}{(n-r)!}\]

This formula is similar to previous one as once r – 1 items have been selected, the number of different choices for the rth item equals n – (r – 1) or n – r +1.

Example:

A candidate for presidential election would like to visit seven cities prior to the next primary election data. But, for him, it will be possible to visit only three of the cities. How many different itineraries (routes, schedules) can he and his staff consider?

Here order of visit is important in planning a itinerary, therefore the number of possible itineraries is equal to the number of permutations of seven cities taken three at a time i.e

\begin{align*}

^nP_r = \frac{n!}{(n-r)!}\\

^7P_3 &= \frac{7!}{(7-3)!}\\

&=\frac{7.6.5.4.3.2.1}{4.3.2.1}\\

&=7.6.5 =210

\end{align*}