Question

Let $C_{1}, \, C_{2}, \, C_{3}...$ are the usual binomial coefficients where $C_{r}= \,{}^{n}C_{r}$ . Let $S=C_{1}+2C_{2}+3C_{3}+...+nC_{n}$ , then $S$ is equal to

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

Answer 1

Answer: (C)

Let $S=C_{1}+2C_{2}+3C_{3}+\ldots +nC_{n}=\Sigma _{\text{r} = 1}^{\text{n}}r\cdot \,{}^{n}C_{r}$
$=\Sigma _{\text{r} = 1}^{\text{n}}r\cdot \frac{n}{r} \,{}^{n - 1}C_{r - 1} \, \, \, \left[\because \,{}^{n} C_{r} = \frac{n}{r} \,{}^{n - 1} C_{r - 1}\right]$
$=n\Sigma _{\text{r} = 1}^{\text{n}} \,{}^{n - 1}C_{r - 1}$
$=n\left[ \,{}^{n - 1} C_{0} + \,{}^{n - 1} C_{1} + \,{}^{n - 1} C_{2} + \ldots + \,{}^{n - 1} C_{n - 1}\right]$
$=n2^{n - 1}$