Is There A Law Of Factorial?

Answer 1

Yes, there is. There is a law of factorial.

1. Law of factorial is confirmed by a general formula to be expressed by words.
2. From there one has a proof without words that zero factorial is equal to one.
3. One has not to accept 0!=1 by convention anymore.
4. And the "0!"is not an empty product.

Author Đoàn Đức Nhuận (in vietnamese)

Explanation of the law
n! = (n+1)n -nn.n +(n -1)n.n(n-1)/2 -(n -2)n.n(n-1)(n-2)/2.3
+ (n -3)n.n(n -1)(n -2)(n -3)/2.3.4 +…
…+(-1)n-4.5 n.n(n -1)(n -2)(n -3)/2.3.4
+(-1)n -3.4n.n(n -1)(n -2)/2.3 +(-1)n -2.3n.n(n -1)/2
+(-1)n -1.2n.n +(-1)n.1
Law of The Factorial's Property

The Factorial of every natural number n is analized and arranged unique by order and is alternately equal to sums and subtractions of the n+1 products of:

every nth exponentiation of a base (from n+1, among n+1 bases, to the last number 1), with

every number (from number 1, anong n+1 numbers, to the last number 1).

Explanation of n!

n,..., n! = (n +1)n - nn.n +… …+(-1)n -1.2n.n +(-1)n.1

……... ... ... ... ... ... ... ...

6,..., 6! = 720 = 1.76 - 6.66 + 15.56 - 20.46 + 15.36 - 6.26 + 1.16

5,..., 5! = 120 = 1.65 - 5.55 + 10.45 - 10.35 + 5.25 - 1.15

4,..., 4! = ..24 = 1.54 - 4.44 + 6.34 - 4.24 + 1.14

3,..., 3! = ....6 = 1.43 - 3.33 + 3.23 - 1.13

2,..., 2! = ....2 = 1.32 - 2.22 + 1.12

1,..., 1! = ....1 = 1.21 - 1.11

0,..., 0! = ...... = 1.10 ................... (1.10 = 1.1 = 1)

One sees from the Explanation of n! that:

All of numbers on rows and columns perform as by law.

From that one has a final logical inductive:

When n = 0, in the column n!, the 0! cannot be equal to any logical value.

But from the Explanation of n!, clearly that one deduces: 0! = 1.10 = 1.1 = 1.

The Factorial has its general property of its own, and from "The Property Formula of Factorial" one proves: 0! = 1.

The thing above is A proof that 0! = 1, isn't it?

And from now on, one has not to accept by convention: 0! = 1.