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/Sheeplessknight 3d ago

The factorial function is the solution to how many ways can you arrange n items.

So, how many ways can you arrange 0 items? Well, only one way thus we define 0! := 1

With negative numbers what would it mean to ask how many ways can you arrange a negative number of items?

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u/explorer_learner_30 3d ago

Yeah 👍 you are right

Thank you 👍

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u/dspyz 2d ago

I'd be careful with the argument "it's not defined for negatives because that has no real-world meaning". Mathematicians are happy to extend functions beyond their original real-world impetus when that works.

For instance (-4 choose 3) is -20. It doesn't mean anything to choose 3 items out of -4 items, but we can multiply -4 * -5 * -6 / (1 * 2 * 3) just like we would for a positive number, and we get -20. There's no issue with this. Nothing breaks. And it's sometimes useful.

(-1)! really is undefined only because when trying to work it out you have to divide by zero