Question

The sum $\frac{1}{1!(n-1)!}+\frac{1}{2!(n-2)!}+\dots \dots \frac{1}{1!(n-1)!}$ is equal to :

• A. $\frac{1}{n!}\left(2^{n-1}-1\right)$
• B. $\frac{2}{n!}\left(2^n-1\right)$
• C. $\frac{2}{n!}\left(2^{n-1}-1\right)$
• D. none

Answer 1

Answer: (C)

$\frac{1}{1!(n-1)!}+\frac{1}{2!(n-2)!}+\frac{1}{1!(n-1)!}\dots +=\frac{1}{n!}\left[{}^n{C}_1+{}^n{C}_2+\dots .+{}^n{C}_{n-1}\right]$ (multiply and divde by $n!$ )
$=\frac{1}{n!}\left[2^n-2\right]=\frac{2}{n!}\left(2^{n-1}-1\right)$