Q. $\frac{1}{1!(n - 1)!} + \frac{1}{3!(n - 3)!} + \frac{1}{5!(n - 5)!} + ...$ is equal to
A) $\frac{2^{n}}{n!}$
B) $\frac{2^{n - 1}}{n!}$
C) 0
D) None of these
Explanation
(b)
Given that,
$\frac{1}{1!(n - 1)!} + \frac{1}{3!(n - 3)!} + \frac{1}{5!(n - 5)!} + ...$
$= \frac{1}{n!}\left\lbrack \frac{n!}{1!(n - 1)!} + \frac{n!}{3!(n - 3)!} + \frac{n!}{5!(n - 5)!} + ... \right\rbrack$
$= \frac{1}{n!}\left\lbrack \ ^{n}C_{1} + \ ^{n}C_{3} + \ ^{n}C_{5} + ... \right\rbrack$
$= \frac{2^{n - 1}}{n!}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.