Combinations Without Repeatition

1. Combinations Without Repeatition refer to Selection of Subsets of Items from a given Set of Distinct Items such that Any One Item can occur Only Once in a Selected Subset/Combination
2. The Count of Combinations Without Repeatition is calculated as follows
   
   Let \(N\) be the Number of Distict Items and \(R\) (where \(R \leq N\)) be the Number of Items that are Selected from \(N\) Items.
   
   Let \(N_C\) be the Total Number of Unique Combinations of \(R\) Items Selected from \(N\) Items.
   
   Also, Linear Permutations Possible for \(R\) Distinct Items = \(R!\)
   
   Now, Number of Linear Permutations Possible for \(1^{st}\) Combination of \(R\) Items Selected from \(N\) Items, \(P_1\)= \(R!\)
   
   Similarly, Number of Linear Permutations Possible for \(2^{nd}\) Combination of \(R\) Items Selected from \(N\) Items, \(P_2\)= \(R!\)
   
   Similarly, Number of Linear Permutations Possible for \(3^{rd}\) Combination of \(R\) Items Selected from \(N\) Items, \(P_3\)= \(R!\)
   \(\vdots\)
   Similarly, Number of Linear Permutations Possible for \({N_C}^{th}\) Combination of \(R\) Items Selected from \(N\) Items, \(P_C\)= \(R!\)
   
   Hence, Total Number of Permutations of \(N\) Items Possible by taking \(R\) Items at a time = \(P_1 + P_2 + \cdots + P_C=N_C \times R!\)   ...(1)
   
   But, as per the formula of Linear Permutations Without Repeatition, Total Number of Permutations of \(N\) Items Possible by taking \(R\) Items at a time= \(\frac{N!}{(N-R)!}\)   ...(2)
   
   From equations (1) and (2) we have
   
   \(N_C \times R!= \frac{N!}{(N-R)!}\hspace{.5cm}\Rightarrow N_C= \frac{N!}{R!(N-R)!}=C(N,R)=C(N,N-R) \)   ...(3)
   
   Equation (3) gives the formula for Combinations Without Repeatition.