Fundamental Principle of Counting, Concept of Factorial, Permutation and Combination

1. Given \(K\) Number of Events \(E_1, E_2, ... , E_k\) that can happen in \(N_1, N_2, ... , N_k\) ways respectively, then as per Fundamental Principle of Counting the Total Number of Ways in which All the Events can happen is
   1. \(N_1 + N_2 + \cdots + N_k\) if the Events are Mutually Exclusive. Also if \(N_1=N_2=\cdots=N_k=N\) then Total Number of Ways in which All the Events can happen is \(K \times N\). This is known as the Sum Rule of Fundamental Principle of Counting.
   2. \(N_1 \times N_2 \times \cdots \times N_k\) if the Events are Not Mutually Exclusive. Also if \(N_1=N_2=\cdots=N_k=N\) then Total Number of Ways in which All the Events can happen is \(N^k\). This is known as the Product Rule of Fundamental Principle of Counting.
2. Given any Integer \(N \geq 0\), the Factorial Function for \(N\) is written as \(N!\) (\(N\) Factorial). This Factorial Function is defined as follows
   
   \(0!=1!=1\)
   
   \(2!=2 \times 1\)
   
   \(3!=3 \times 2 \times 1\)
   \(\vdots\)
   \(N!=N \times (N-1) \times (N-2) \times \cdots \times 2 \times 1\)
3. The Fundamental Principle of Counting and the Factorial Function are primarily used for following 2 purposes
   1. Calculating Number of Ways Possible to Select Objects from a Given Set of Objects. Each such Unique Selection is called a Combination.
   2. Calculating Number of ways Possible to Arrange Objects across Any Particular Symmetry from a Given Set of Objects. Each such Unique Arrangement is called a Permutation.
4. Both Combinations and Permutations can be done either Without Repeatition, With Unrestricted Repeatition or With Restricted Repeatition from a Given Set of Distinct Objects. Whereas Combinations involve selecting the Objects, Permutations involve Arranging the Objects across a Particular Symmetry. The 2 most Commonly Studied Symmetries accross which Items are arranged are Linear (which involves arranging Selected Items in a Line) and Circular (which involves arranging Selected Items in a Circle) Symmetries.
   
   The following gives the list of All Kinds of Possible Combinations and All Kinds of Possible Permutations accross Linear and Circular Symmetries (without involving any other conditions) and Corresponding Formulae to Calculate the Count of them.
   
   

   S.No.	Permutation Type/Combination	Count Without Repeatition	Count With Unrestricted Repetition	Count With Restricted Repetition
   1.	Linear Permutation	\(N!\)	\(N^R\)	\(\frac{N!}{P!Q!R!}\)
   2.	Direction Independent Linear Permutation	\(\frac{N!}{2}\)	\(\frac{N^R + N^{\frac{R}{2}}}{2}\) if \(R\) is Even \(\frac{N^R + N^{\frac{R+1}{2}}}{2}\) if \(R\) is Odd	Complex Formula
   3.	Linear Permutation of Subsets	\(P(N,R)=\frac{N!}{(N-R)!}\)	-	Calculated Using Generating Functions
   4.	Direction Independent Linear Permutation of Subsets	\(\frac{P(N,R)}{2}=\frac{N!}{2(N-R)!}\)	-	Complex Formula
   5.	Circular Permutation	\((N-1)!\)	\(\frac{N^R + (R-1)N + (\sum_{i=1}^n(R- F_i)\times F_{i}Count(C_{NF_i}))}{R}\)	Complex Formula
   6.	Orientation Independent Circular Permutation / Necklace Permutation	\(\frac{(N-1)!}{2}\)	Complex Formula	Complex Formula
   7.	Circular Permutation of Subsets	\(\frac{P(N,R)}{R}=\frac{N!}{R(N-R)!}\)	-	Complex Formula
   8.	Orientation Independent Circular Permutation of Subsets	\(\frac{P(N,R)}{2R}=\frac{N!}{2R(N-R)!}\)	-	Complex Formula
   9.	Combination	\(C(N,R)=C(N,N-R)=\frac{P(N,R)}{R!}=\frac{N!}{R!(N-R)!}\)	\(C(N+R-1,R)=C(N+R-1,N-1)\)	Calculated Using Generating Functions