1) The factorial function (n!) is defined as the product of all positive integers from 1 to n. Using this definition, 1! equals 1 as it is 1 x 1. 2) Logically, n! can be expressed as n x (n-1)!. Therefore, 1! equals 1 x 0! which simplifies to 1 = 0!. 3) Factorials represent the number of permutations of arranging a set of numbers. An empty set containing 0 numbers can only be arranged one way, so 0! equals 1.

1. Why 0!=1
Example (1)
If n! is defined as the product of all positive integers from 1 to n, then:
1! = 1×1 = 1
2! = 1×2 = 2
3! = 1×2×3 = 6
4! = 1 × 2 × 3 × 4 = 24
...
𝑛! = 1 × 2 × 3 ×. . .× (𝑛 − 2) × (𝑛 − 1) × 𝑛
and so on.
Logically, n! can also be expressed 𝑛 × (𝑛 − 1)! .
Therefore, at n=1, using 𝑛! = 𝑛 × (𝑛 − 1)!
1! = 1 × 0!
which simplifies to 1 = 0!

2. Example (2)
Why 0!=1
The idea of the factorial (in simple terms) is used to compute the number of permutations of
arranging a set of n numbers.
n Number of Permutations (n!) Visual example
1 1 {1}
2 2 {1,2}, {2,1}
3 6 {1,2,3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, {3,2,1}
Therefore,
0 1 {}
It can be said that an empty set can only be ordered one way, so
0! = 1.