Linear Permutations With Unrestricted Repeatition

1. Linear Permutations With Unrestricted Repeatition refer to All Distinct Linear Arrangements of \(N\) Number of Distinct Type of Items in a sequence of a given Length \(R\), such that Any Item can occur any number of times (between \(0\) and \(R\)) in any given Linear Arrangement.
2. Since the \(R\) Number of Positions can be occupied by \(N\) Number of Distinct Type of Items, each of which can get repeated any number of times (between \(0\) and \(R\)) in a particular arrangement, the Count of Total Number of Linear Permutations With Unrestricted Repeatition can be calculated as follows
   
   The \(1^{st}\) Position can be occupied by any of \(N\) Possible Items
   
   The \(2^{nd}\) Position can also be occupied by any of \(N\) Possible Items
   
   The \(3^{rd}\) Position can also be occupied by any of \(N\) Possible Items
   \(\vdots\)
   The \(R^{th}\) Position can also be occupied by any of \(N\) Possible Items
   
   Therefore as per Product Rule of Fundamental Principle of Counting, the Count of Total Number of Possible Arangements are
   
   \(N \times N \times N \times\) (Repeated \(R\) times)= \(N^R\)
3. The following are 2 examples of Linear Permutations With Unrestricted Repeatition
   1. The total number of 3 Letter Words that can be formed by using 26 characters of English Alphabets are \({26}^3\).
   2. The total number of 5 Digit Numbers that can be formed by using the 10 digits \(0\) to \(9\) are \({10}^5\).
4. Since the Items can get Repeated in an Arrangement, some of the Permutations are Symmetric / Palindromic while others are Asymmetric / Non-Palindromic. The Total Number of Symmetric Linear Permutations out of \(N^R\) possible Permutations are calculated as
   
   When \(R\) is Even= \(N^{\frac{R}{2}}\)
   
   When \(R\) is Odd = \(N^{\frac{R+1}{2}}\)
   
   Hence, the Total Number of Asymmetric Linear Permutations out of \(N^R\) possible Permutations are calculated as
   
   When \(R\) is Even = \(N^R - N^{\frac{R}{2}}\)
   
   When \(R\) is Odd = \(N^R - N^{\frac{R+1}{2}}\)
5. The Total Number of Direction Independent Linear Permutations out of \(N^R\) possible Permutations are calculated as
   
   Count of Direction Independent Linear Permutations= \(\frac{No.\hspace{.2cm}of\hspace{.2cm}Asymmetric\hspace{.2cm}Permutations}{2} + No.\hspace{.2cm}of\hspace{.2cm}Symmetric\hspace{.2cm}Permutations\)
   
   Therefore Count of Direction Independent Linear Permutations when \(R\) is Even = \(\frac{N^R - N^{\frac{R}{2}}}{2} + N^{\frac{R}{2}} = \frac{N^R + N^{\frac{R}{2}}}{2}\)
   
   And Count of Direction Independent Linear Permutations when \(R\) is Odd = \(\frac{N^R - N^{\frac{R+1}{2}}}{2} + N^{\frac{R+1}{2}} = \frac{N^R + N^{\frac{R+1}{2}}}{2}\)