Linear Permutations With Restricted Repeatition refer to Linear Arrangements of \(K\) Number of Distinct Type of Items
in a sequence of a given Length \(N\) (where \(N > K\)), such that All Items Types occur at least Once and Some or All Item Types can occur More than Once (upto a Maximum Fixed Number for Each Item Type) in any given Linear Arrangement.
The Count of Linear Permutations With Restricted Repeatition is calculated as follows
Number of Distinct Type of Items = \(K\)
Length of the Each Arrangement = \(N\) (where \(N>K\))
Number of ways \(N\) Items can be arranged = \(N!\)
Since \(N > K\), therefore some of the Items are repeated in the arrangements. Let consider any Item \(A\) that gets repeated \(P\) times in any arrangement of the \(N\) Items.
Number of ways \(P\) Number of Item \(A\) can be arranged = \(P!\)
Since for any given arrangement of \(N\) Items there are \(P!\) Number of ways \(P\) Number of Item \(A\) can be arranged, therefore Number of Permutations of \(N\) Items that get merged into a Single Permutation is also \(P!\) (as they are indistinguishable from each other).
Hence, Net Total Number of Permutations With Restricted Repeatition of Item \(A\)= \(\frac{Total\hspace{.1cm}Number\hspace{.1cm}of\hspace{.1cm}Permutations\hspace{.1cm}of\hspace{.1cm}N\hspace{.1cm}Items}
{Total\hspace{.1cm}Number\hspace{.1cm}of\hspace{.1cm}Permutations\hspace{.1cm}Getting\hspace{.1cm}Merged\hspace{.1cm}into\hspace{.1cm}Single\hspace{.1cm}Permutation\hspace{.1cm}Because\hspace{.1cm}of\hspace{.1cm}Repeatition\hspace{.1cm}of\hspace{.1cm}Item\hspace{.1cm}A}=\frac{N!}{P!}\)
Now, along with Item \(A\), lets consider 2 more Items \(B\) and \(C\) that get repeated \(Q\) and \(R\) times respectively in any arrangement of the \(N\) Items.
Number of ways \(Q\) Number of Item \(B\) can be arranged = \(Q!\)
Number of ways \(R\) Number of Item \(C\) can be arranged = \(R!\)
Therefore as per Product Rule of Fundamental Principle of Counting, the Total Number of Arrangements Possible of Items \(A\), \(B\) and \(C\) taken together are = \(P!\times Q!\times R! = P!Q!R!\)
Since for any given arrangement of \(N\) Items there are \(P!Q!R!\) Number of ways \(P\) Number of Item \(A\), \(Q\) Number of Item \(B\) and \(R\) Number of Item \(C\) can be arranged, therefore Number of Permutations of \(N\) Items that get merged into a Single Permutation is also \(P!Q!R!\) (as they are indistinguishable from each other).
Hence, Net Total Number of Permutations With Restricted Repeatition of Items \(A\), \(B\) and \(C\)