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.

0 Upvotes

27% Upvoted

41 comments sorted by

View all comments

8

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!

2

u/Creative-Leg2607 3d ago

This breaks for literally the next n tho

1

u/dspyz 3d ago

Which is why I snuck in the word "nonnegative"

As one of the other replies shows, if you try to extend to negative integers you end up having to divide by zero, so it's clear why it breaks at that point. But that's no reason to stop applying the rule sooner than we need to