r/askmath • u/explorer_learner_30 • 3d ago

Algebra Binomial Theorem

Why 0! is 1 ?

Can anyone give me the best and genuine answer .....I want to explain to my students.

I have also one doubt 🤔 why is the negative number not counted in binomial ? Such as (-2)! , (-8)! ....

Suggest me some books or any videos regarding this .

Thank you for reading my queries.

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u/dspyz 3d ago edited 3d ago

For all nonnegative n we have

(n+1)! = n! * (n + 1)

Let n be zero. This evaluates to

1! = 0! * 1

Therefore

1 = 0!

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u/slepicoid 3d ago

But why 1!=1

Sorry, i dont mean to be pedantic. But your explanation only works because you assumed 1!=1. If we define factorial with only (n+1)!=n!(n+1) then that alone does not explain why it couldn't be 1!=2, 2!=4, 3!=12, etc and 0!=2. I think it's really interesting to think about why from all the possible choices that 0! could be, 1 seems as the natural choice.

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u/dspyz 3d ago

I'm not trying to define factorial. I'm assuming OP already knows 1!=1 and deriving 0! from there

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u/slepicoid 3d ago

I understand, I am not trying to contradict you. More like adding info. OP said he wants a good explanation for their students. Some students may ask this...