Question

$1{\cdot }^n{\text{C}}_1+$ $2{\cdot }^n{\text{C}}_2+$ $3{\cdot }^n{\text{C}}_3+...+$ ${}^n{\text{C}}_n$ is equal to

• A. $\frac{n(n+1)}{4}{.2}^n$
• B. $2n^{n+1}-3$
• C. $n{.2}^{n-1}$
• D. none of these

Answer 1

The correct option is C $n{.2}^{n-1}$

Above series can be expressed as
$={\sum }_{r=1}^{n}r{}^n{C}_r$
$={\frac{d(1+x{)}^n}{dx}}_{x=1}$
$=n(1+x{)}_{x=1}^{n-1}$
$=n{2}^{n-1}$