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.
When considering the numbers of different groups which may be formed from a collection of objects one frequently finds the need to make calculations of the form, 4 x 3 x 2 x 1 for 4 objects. I suppose I must explain why that is.
Say we have the four objects A, B, C, D.
If I select from that collection to form another group, you will see that I
could choose any one of the four for the first selection. This
would leave three objects. I could select any of those three for the second selection,
leaving any one of two for the next and so on. The total number of ways I could select
those objects would be 4 x 3 x 2 x ... , and the series would stop at one when I selected
the last object. To save repeatedly writing down long strings of such products the
notation 4! is used to represent 4 x 3 x 2 x 1 or 6! for 6 x 5 x 4 x 3 x 2 x 1.
The ! is read as factorial. So the examples quoted above are more easily written as 4!
and 6!. If you care to calculate their values they are 24 and 720 respectively.
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.
Why not tell your friends about it? Help dispel the widespread ignorance about 0!