Question

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

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

Answer 1

Answer: (C)

Since $r{\cdot }^n{C}_r=n{\cdot }^{n-1}{C}_{r-1}$
$\therefore$ reqd. sum
$=n\left[{}^{n-1}{C}_0+{}^{n-1}{C}_1+{}^{n-1}{C}_2+..........+{}^{n-1}{C}_{n-1}\right]$
$=n\cdot 2^{n-1}$