why the factorial of 0 is 1 ?

Answer 1

PERMUTATIONAL VIEW.
the number of ways of arranging N objects in N places is N!(hope u know ! means factorial.)therefore the number of ways of arranging 0 objects in N places(=not placing any object in any of the N places) is also one way.

also, one can think of using a COMBINATIONS.

Answer 2

Usually n factorial is defined in the following way:

n! = 1*2*3*...*n
But this definition does not give a value for 0 factorial, so a natural question is: what is the value here of 0! ?

A first way to see that 0! = 1 is by working backward. We know that:

1! = 1
2! = 1!*2
2! = 2
3! = 2!*3
3! = 6
4! = 3!*4
4! = 24


We can turn this around:
4! = 24
3! = 4!/4
3! = 6
2! = 3!/3
2! = 2
1! = 2!/2
1! = 1
0! = 1!/1
0! = 1


In this way a reasonable value for 0! can be found.
How can we fit 0! = 1 into a definition for n! ? Let's rewrite the usual definition with recurrence:

1! = 1
n! = n*(n-1)! for n > 1


Now it is simple to change the definition to include 0! :
0! = 1
n! = n*(n-1)! for n > 0


Why is it important to compute 0! ?
An important application of factorials is the computation of number combinations:

n!
C(n,k) = --------
k!(n-k)!


C(n,k) is the number of combinations you can make of k objects out of a given set of n objects. We see that C(n,0) and C(n,n) should be equal to 1, but they require that 0! be used.
n!
C(n,0) = C(n,n) = ----
n!0!


So 0! = 1 neatly fits what we expect C(n,0) and C(n,n) to be.
Can factorials also be computed for non-integer numbers? Yes, there is a famous function, the gamma function G(z), which extends factorials to real and even complex numbers. The definition of this function, however, is not simple:

inf.
G(z) = INT x^(z-1) e^(-x) dx
0


Note that the extension of n! by G(z) is not what you might think: when n is a natural number, then G(n) = (n-1)!
The gamma function is undefined for zero and negative integers, from which we can conclude that factorials of negative integers do not exist.

Answer 3

Mr. Sumans answer is laudable. To make his view complete I would like to add a little to make his view complete.
True to say, Maths is proud of its straightforwardness, truthfulness.And it is well known for its honesty and punctuality in following its own principles.
But, at least in sporadic cases every body and every thing need to compromise with the nature to retain its existence and make it usable also continuous.
The above question is one such case.
For example when n-n factorial is not equal to 1, the expression becomes non-pragmatic (impractical). Hence to make it practicable it is considered that 0 factorial is 1
So in applied mathematics 0 ! is 1
Where as in pure mathematics it is not possible to agree this statement.

Answer 4

"Factorial of zero is one" is equally paradixical as "square root of minus one"(imaginary)!

It is a simple fact that ancient Indians used 'a digital before-units counting' to carry zero into number applications (say 1 or,2 or,3 or,4or ... any higher digital number position have respective 'a-unit-less' before unit conditions like 0 or, 1 or, or 2, or 3...or (any higher digital number position-1)!

Unfortunately "a-unit-less" before unit conditions emerged well over 5000 years ago and 'the concept' disappeared leaving 'zero' with Indians!

Zero went to Europe (via Arabia) and a lot of strange 'numbers less than zero' serrounded it. Since then zero lost its "absolute nothingness-sense" and infinitesimal value started to rule mathematics!

Any body who recognize an absolute-zero-state, ( which is possible by a before units/unit counting alone ) will agree that a conceptual error exists in between 'ancient Indian zero' and 'current zero'! People need zero even today but not an Indian-concept!

Today, computers accurately use zero and people don't need said computer accuracy! An infinitesimal value and related complications fullfill human computation needs!

Computing principles like ( square root of "a-unit less" than zero) which is "imaginary" will be used by people endlessly!

Apparently "factorial of zero is one" is one of those computing principles originated by those who do not admit that "zero is a digital whole number like "one" or an "any number of a-unit merged number"

Iam too happy to be an Indian fool!

Regards.

Answer 5

it is defined for easy compuataion and avoid boundary cases,

please refer to

Answer 6

it is defined as one