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Mark the correct alternative in each of the following:
The coefficient of \( \frac{1}{x} \) in the expansion of \( (1+x)^{n}\left(1+\frac{1}{x}\right)^{n} \) is
(A) \( \frac{n !}{\lfloor(n-1) !(n+1) !\}} \)
(B) \( \frac{(2 n) !}{[(n-1) !(n+1) !]} \)
(C) \( \frac{(2 n) !}{(2 n-1) !(2 n+1) !} \)
(D) none of these
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