A) \[\frac{{{2}^{n}}}{n!}\]
B) \[\frac{{{2}^{n-1}}}{n!}\]
C) 0
D) none of these
Correct Answer: B
Solution :
\[\frac{1}{1!(n-1)!}+\frac{1}{3!(n-3)!}+\frac{1}{5!(n-5)!}+....\] Multiplying numerator and denominator by \[n!\] \[=\frac{1}{n!}\left[ \frac{n!}{1!(n-1)!}+\frac{n!}{3!(n-3)!}+\frac{n!}{5!(n-5)!}+... \right]\] \[=\frac{1}{n!}\left[ {{\,}^{n}}{{C}_{1}}{{+}^{n}}{{C}_{3}}{{+}^{n}}{{C}_{5}}+... \right]\] \[=\frac{1}{n!}{{2}^{n-1}}\]
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