Linear Permutations

Introduction 

When objects are arranged in a straight line, the different orders in which they are arranged are called linear permutations.

For example, four boys and three girls are standing in a straight line. The different orders in which they can stand are represented below: 

• BBBBGGG
• BGBGBGB
• GGGBBBB
• ... and many more 

All of these different orders are called linear permutations.

Formulas

Linear permutations are calculated by different formulas, depending on the type of objects which are arranged in a straight line.

When all objects are different (unique from each other), and all objects are arranged in different orders, linear permutations of these objects are given by: 

`P = n!`

(The above result follows directly from the Fundamental Principle of Counting)

The above formula is explained in more detail in this post.

When all objects are different (unique from each other) and only some of the objects are taken at a time and arranged in different orders, the linear permutations of these objects are given by:

`P_r^n = (n!)/(n-r)!`

(the above formula gives the total number of different orders, that is, different permutations, in which 'n' different objects can be arranged, taken 'r' at a time)

The above formula is explained in more detail in this post.

When there are 'n' objects, out of which 'p' are same and of one kind, 'q' are the same and of a different kind, and 'r' are also same and of a different kind, then the total number of ways in which the 'n' objects can be arranged in different orders are

`P = (n!)/(p!q!r!)`

(in the above formula, the 'p', 'q', and 'r' objects together make the total of 'n' objects. That is, p + q + r = n)

The above formula is explained in more detail in this post.

When there are 'n' objects, alike in all respects, then any order in which they are arranged is the same. Thus, the number of permutations of 'n' alike objects is 1.