Factorial of zero

[shahar]

I see a question that ask: "What are the uses of 0!" or "Why are need that !0 = 1 in calculations?
I thought to share the question with you.
Why 0! = 1?
How does it help us in calculations?

 

[fresh_42]

I see a question that ask: "What are the uses of 0!" or "Why are need that !0 = 1 in calculations?
I thought to share the question with you.
Why 0! = 1?
How does it help us in calculations?

It makes expressions like [imath] e^x=\displaystyle{\sum_{k=0}^\infty \dfrac{x^k}{k!}} [/imath]
a natural notation.

My opinion is that the faculty is a multiplicative operation by definition. Hence, [imath] 0! [/imath] should equal the neutral element of multiplication, which is [imath] 1. [/imath] You can likewise argue that [imath] 0! [/imath] is an empty product, and therefore equals [imath] 1, [/imath] just as empty sums equal [imath] 0. [/imath] This handling of products makes notation and therewith calculations easier.

 

[Dr.Peterson]

I see a question that ask: "What are the uses of 0!" or "Why are need that !0 = 1 in calculations?
I thought to share the question with you.
Why 0! = 1?
How does it help us in calculations?

We use 0! in permutations and combinations; for example, [imath]_n\text{C}_n=\frac{n!}{n!0!}=1[/imath], which would be wrong if 0! had any other value than 1.

Similarly, [imath]_n\text{P}_n=\frac{n!}{0!}=n![/imath].

So the main value of this definition is that it makes all the formulas that use factorials consistent, without needing special treatment for zero.

Not to mention that it makes sense, as @fresh_42 says. As another example, observe that [imath]n!/n=(n-1)![/imath], and for [imath]n=1[/imath], this implies that [imath]1!/1=(1-1)!=0![/imath], and this has to be 1.

 

[kurksai]

I see a question that ask: "What are the uses of 0!" or "Why are need that !0 = 1 in calculations?
I thought to share the question with you.
Why 0! = 1?
How does it help us in calculations?










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Great question. The easiest way to see it is just by following the math pattern backwards:

3! = 6

2! = 6 / 3 = 2

1! = 2 / 2 = 1

Therefore, 0! = 1 / 1 = 1.

Conceptually, a factorial just asks "how many ways can you arrange X things?" How many ways can you arrange zero things on a table? Exactly 1 way: by doing nothing.

As for why it's useful: if 0! was 0, it would break a ton of formulas in probability and algebra. Equations for combinations and permutations have factorials in the denominator. If 0! = 0, you would constantly end up dividing by zero and breaking the math. Defining it as 1 keeps everything working perfectly without needing a bunch of messy exceptions.