Introduction to factorial notation

Definition

• Let n ∈ 𝕎, the set of whole numbers the symbol “n!” is read as “n factorial”.
• Let n be a natural number, then n! = 1 × 2 × 3 × … × n (the product of the first n-natural numbers)

Note

0! = 1
If n is a positive integer, then n! = n(n − 1)!

Permutations

Definition

• A permutation is an ordered arrangement of sequence of all or part of a finite set of things or objects. In permutation, order of objects is very important.
• If the objects are arranged in a line, then the permutation is said to be linear permutation or simply permutation,
• If the objects are arranged in the form of a circle, then the permutation is said to be circular permutation.
• If there are n objects, then the number of arrangements that can be formed by taking r objects chosen from a set of n different objects is
  

Verification (Illustration)

If there are r labels and n different objects, then we can have the following techniques of labeling the n objects in the r labels.

1. The number of Permutations of N objects, P of which are identical is

2. The number of permutations of N objects, P of which are identical and Q of which are identical (but different from the set of P objects) is

3. The number of permutations of N objects n_j of which are identical, where j = 1, 2, 3, 4, 5, 6, 7, 8, 9, . . . , k
N = n₁ + n₂ +… + n_k and the set of n_j objects is different from the set of n_i objects for all