Question

The value of $\frac{1}{(n-1)!}+\frac{1}{(n-3)!3!}+\frac{1}{(n-5)!5!}+....$

• A. $\frac{2^{n+1}}{n!}$
• B. $\frac{2^{n-1}}{n!}$
• C. $\frac{2^{n-1}}{n+1!}$
• D. $\frac{2^{n+1}}{n-1!}$

Answer 1

The correct option is B $\frac{2^{n-1}}{n!}$

$\frac{1}{(n-1)!}+\frac{1}{(n-3)!3!}+\frac{1}{(n-5)!5!}+....$
$=\frac{1}{n!}(\frac{n!}{(n-1)!1!}+\frac{n!}{(n-3)!3!}+\frac{n!}{(n-5)!5!}+....)$
$=\frac{1}{n!}{(}^n{C}_1{+}^n{C}_3{+}^n{C}_5+.............)=\frac{2^{n-1}}{n!}$