Zero does not = 1, but why does zero factorial = 1 factorial? NEED A SIMPLE STEP BY STEP EXPLANATION, PLEASE?

Question

DH has tried to explain this repeatedly, but with no success (it's too much like a "crooked bellboy" problem, and my math phobia kicks in). Anyone out there want to do a lot of work for 2 lousy points, keeping in mind that you'll have to keep it at about a grade school level so I can understand it? Or point me towards a site that can go verrrrryyyyy sllooooooowwwly? Thanks!

Answer 1

1. How may ways are there to arrange 2 balls, one is orange and another is red in color?

Orange, Red
Red, Orange

2 ways.

2. How may ways are there to arrange 1 ball?

1 way

3. How many ways are there to arrange 0 ball? To put it in another way, how many ways are there NOT to arrange the ball.

1 way

Answer 2

Simple answer: 0! (read "Zero Factorial") is defined to equal 1.

Involved answer(s):
There are several proofs that have been offered to support this common definition.

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!

Answer 3

The answer lies in 1!
1! = 1
and 1! = 0! * 1
so 0! has to be 1 because only 1*1 = 1;
else every other factorial would be zero or there wouldnt be any other factorial and you wouldnt have asked this question.
so 0! is assumed to be 1.

Answer 4

It's very simply, it was a rule made up so that when used in factorial calculations, the answers would work out.
It's simply a rule. Play by the rules and we should all come up with the same answer for a given question.
0! = 1.

The bill was put to the house of commons and every one say,
'here here'. From that day to this that's the way it's been. ☺

Answer 5

For all nonnegitive intergers n, n! is defined recursively as follows:

0! = 1

1! = 1

(n + 1)! = n!(n +1), n ≥1

Note:

0! = 1 and 1! = 1 by definition.



Hey Doug:

Γ(n + 1) = n! for every nonnegitive integer n.

He said grade school level, not grad school. I don't think he's got the math background for calculus or differential equations.

Answer 6

The factorial function is formally defined by

n! =
n
∏ k for all n ≥ 0
k=1
The above definition incorporates the convention that
0! = 1
as an instance of the convention that the product of no numbers at all is 1. This fact for factorials is useful, because

the recursive relation (n + 1)! = n! × (n + 1) works for n = 0;
this definition makes many identities in combinatorics valid for zero sizes.
In particular, the number of arranging or permutations of an empty set is in just one way.

Answer 7

cuz..factorial is used in calculatin the number of ways for something to happen..i.e. 4!,number of ways for 4 person to sit on 4 chairs in a line..
_ _ _ _..as ya can see..the first chair from the left can have 4 person choosin it..then the next is 3..then 2..then finally one..therefore it is 4*3*2*1 = 24..! is juz a notation else wt would have happened if there was 72!..as for 0!..it means the number of ways which zero people can sit on zero chairs which is one..get it?..it has nothin to do with 0 = 1..or wtever..1! means the number of ways for 1 guy sit on 1 chair..which is 1..

Answer 8

It's a convention, but it's a very useful one. Just like saying that 0/0 is undefined.

There **is** a function called the 'Gamma' function which, for integer arguments (n), has as values the value of n! and it's value at 0 is one. But that's a function that isn't usually seen until your first year as a grad student in mathematics.


Doug

Answer 9

How many ways can you arrange 0 objects?

only 1 ;)

So for reasons related to permutations and combinations 0! = 1


EDIT

Doug... :P gamma(n) = (n-1)!

Why do people make that mistake so often

Answer 10

You don't have a proof for this.

Get this:

Zero Factorial IS DEFINED as ONE.

that's that.