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Comments
the answer to this problem can be found by the gamma function which is defined as the integration of (e^-x).(x^m-1) where x varies from 0 to infinity; results in gamma m (m>0).. which is also equivalent to (m-1)! putting m=1 and solving the integration we get 0! = 1 ! abhhi bhool jaa engineering mein jayega tab indirectly iska proof mil jayega .. !! :)
Ultimately it's all just a convention. When 0!=1, many things are simplified; for example, the binomial formula (kn)=k!(n−k)!n! holds true even for k=0,n, and it follows naturally from the identity n!=(n−1)!⋅n for n=1. You can freely define it otherwise, but things become more complicated then (for example, the binomial formula works "for 1≤k≤n−1 only; if k=0,n, the result is 1").
The same thing applies for, for example, x0=1 for nonzero x, or that 1 is not a prime number, or that 0 is even. They are just definitions and you're free to change them, but they generally make things more complicated to state if their definitions are changed.
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
*italics*
or_italics_
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[example link](https://brilliant.org)
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or\[
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to ensure proper formatting.2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
Comments
the answer to this problem can be found by the gamma function which is defined as the integration of (e^-x).(x^m-1) where x varies from 0 to infinity; results in gamma m (m>0).. which is also equivalent to (m-1)! putting m=1 and solving the integration we get 0! = 1 ! abhhi bhool jaa engineering mein jayega tab indirectly iska proof mil jayega .. !! :)
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Thanks
One way to explain this is similar to a way to explain why x0=1 (x=0):
We know that n!=n(n−1)! 1!=1(0)! 0!=1
Ultimately it's all just a convention. When 0!=1, many things are simplified; for example, the binomial formula (kn)=k!(n−k)!n! holds true even for k=0,n, and it follows naturally from the identity n!=(n−1)!⋅n for n=1. You can freely define it otherwise, but things become more complicated then (for example, the binomial formula works "for 1≤k≤n−1 only; if k=0,n, the result is 1").
The same thing applies for, for example, x0=1 for nonzero x, or that 1 is not a prime number, or that 0 is even. They are just definitions and you're free to change them, but they generally make things more complicated to state if their definitions are changed.