Can we prove that 0! = 1?
Also what is 0^{0}?

Can we prove that 0! = 1?
Also what is 0^{0}?
I don't think one can "prove" 0! = 1 because that is a matter of definition of the factorial. However, 0! = 1 is necessary for many combinatoric expressions to work. Also, if one generalises the factorial to nonintegers using the gamma function, then 0! = 1 is a natural result.
An indeterminate form.
Lol, the second equation assumes, that 0! = 1.
So can we really use it to prove the former.
No. In the definition of you provided, is axiomatic (part of the definition of ). The second part of the definition is a recursive definition for all integers greater than zero. Being recursive, it requires an initial condition to start from (the first part of the definition). In other words, both parts are necessary to define . Therefore, the second part cannot prove the first part.
n!=n(n1)!
=> 1!=1*0!
but 1!=1
so 0!=1
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