belive it or not !!!

Why should you be concerned that 0!=1? You may not even know what a factorial number is.

Reading the following you may come to understand the idea of a factorial. You may also be able to please your friends and confound your enemies by being able to show that 0!=1. Here is an explanation, requiring only knowledge of simple arithmetic to understand.

In general any factorial number (call it n!), may be written,
n! = n x (n-1) x (n-2) x (n-3) x ... x 2 x 1

This is the general definition of a factorial number.

If you want it in words; a factorial number is the product of all positive integers from 1 to the number under consideration.

The main place it is likely to be encountered is when considering those groups and arrangements of objects mentioned above.

So where does all this 0! stuff fit in?

Nobody has trouble in stating 2! = 2 x 1 , or even that 1! = 1, but 0! appears to make no sense.

It does however, have a value of 1. This is rather counter intuitive but arises directly from our general definition.
n! = n x (n-1) x (n-2) x (n-3) x ... x 2 x 1

Notice this may be written,
n! = n x (n-1)! Still exactly the same definition.

If the left hand side (LHS) = the right hand side (RHS) then dividing both sides by n should leave them still equal, so it is still true to write,
n!/n = n x (n-1)!/n

The (n-1)! in the RHS is being both multiplied by n and divided by n. These cancel leaving,
n!/n = (n-1)! If you doubt this, try it with real numbers, e.g. 4!/4 = 3! or (4 x 3 x 2 x 1)/4 = 3 x 2 x 1 = 6

The equation we now have is,
n!/n = (n-1)!

It is still our original definition in a rearranged form. For convenience I shall write it the other way round.
(n-1)! = n!/n

We also said that our factorial uses the positive integers 1 and above. Try the value of n=2 in our rearranged formula and we get,
(2-1)! = 2!/2 or 1! = 2x1/2

The RHS calculates to 1, so we have the statement 1!=1 That is what we guessed intuitively above. It is now confirmed. But look what happens when we substitute the legitimate value of n=1 in our formula.
(1-1)! = 1!/1

Evaluating this statement gives
0! = 1!/1

We have just shown 1!=1 so the RHS is 1/1 or 1.

Our rearranged definition of a factorial number gives directly the statement,
0!=1 Counter intuitive perhaps but if the definition is true then this is true.

Why not tell your friends about it? Help dispel the widespread ignorance about 0!