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 a Subset of Objects from a Given Set of Objects. Each such Unique Selection is called a Combination.
   2. Calculating Number of ways Possible to Arrange Subset of Objects 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 or With Repeatition from a Given Set of Distinct Objects.
5. Permutations can be done either in a Symmetry Independent Manner or Across a Particular Symmetry.
6. The following gives the list of Formulae/Methods used to Calculate the Count of Combinations and Permutations.
   
   In the table below, \(N\) is the Number of Unique Objects and \(R\) is the Length of Selection/Arrangement.
   
   

   S.No.	Type	Count Without Repeatition (only when \(N \geq R\))	Count With Repetition
   1.	Combination	\(C(N,R)=C(N,N-R)\) \(=\) \({\Large\frac{P(N,R)}{R!}}={\Large\frac{N!}{R!(N-R)!}}\)	\(C(N-1+R,R)\) \(=\) \(C(N-1+R,N-1)\) \(=\) \({\Large\frac{(N-1+R)!}{R!(N-1+R)!}}\)
   2.	Symmetry Independent Permutation	\(P(N,R)={\Large\frac{N!}{(N-R)!}}\) (\(N!\) when \(N=R\))	\(N^R\) Calculated Using Generating Functions for Multinomial Expansion
   2.	Linear Symmetry Permutation	\({\Large\frac{P(N,R)}{2}}={\Large\frac{N!}{2(N-R)!}}\) (\({\Large\frac{N!}{2}}\) when \(N=R\) )	\(\frac{N^R + N^{\frac{R}{2}}}{2}\) if \(R\) is Even \(\frac{N^R + N^{\frac{R+1}{2}}}{2}\) if \(R\) is Odd Calculated Using Linear Symmetry Generating Functions
   4.	Circular Symmetry Permutation	\({\Large\frac{P(N,R)}{R}}={\Large\frac{N!}{R(N-R)!}}\) (\((N-1)!\) when \(N=R\))	Calculated Using Rotational Symmetry Generating Functions
   5.	Necklace Symmetry Permutation	\({\Large\frac{P(N,R)}{2R}}={\Large\frac{N!}{2R(N-R)!}}\) (\({\Large \frac{(N-1)!}{2}}\) when \(N=R\))	Calculated Using Rotational & Reflectional Symmetry Generating Functions