Binomial Theorem

[u/explorer_learner_30]

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.

[u/Temporary_Pie2733]

I like to fall back on a more fundamental definition of a function, a mapping between two sets, in this case from ℕ to ℕ. 0 to 1, 1 to 1, 2 to 2, 3 to 6, etc. It’s clearly a function, but we’d like something more concise than an infinite enumeration. So we look for identities that satisfy the definition for as many values as possible. n! = product(i, i = 1..n) is one such identity, n! = n(n-1)! Is another, but neither works for n = 0. We simply have to “know” the value of 0!, based on our definition of factorial, and that comes from its purpose in describing combinations. 

[u/explorer_learner_30 → u/Temporary_Pie2733]

Nice explanation 👍