Question

If $C_0,C_1,C_2,\dots ,C_n$ denote the binomial coefficients in the expansion of $(1+x{)}^n,$ then value of $1^2\cdot C_1+2^2\cdot C_2+3^2\cdot C_3+\dots +n^2{C}_n$ is

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

Answer 1

The correct option is B $n(n+1)2^{n-2}$

$S_n=1^2\cdot C_1+2^2\cdot C_2+3^2\cdot C_3+\cdots \cdot +n^2\cdot C_n$

$=\sum _{r=1}^n{r}^2\cdot C_r$ where $C_r={}^n{C}_r$

$=\sum _{r=1}^n\left(r\right(r-1)+r)\cdot C_r$

$=\sum _{r=1}^{n}r(r-1)\cdot {}^n{C}_r+\sum _{r=1}^n\cdot r\cdot {}^n{C}_r$

$[\because \frac{{}^n{C}_r}{{}^{n-1}{C}_{r-1}}=\frac{n}{r}]$

$=\sum _{r=2}^{n}r(r-1){}^n{C}_r+n\sum _{r=1}^n{}^{n-1}{C}_{r-1}$


$=n(n-1)\sum _{r=2}^n{}^{n-2}{C}_{r-2}+n\cdot \sum _{r=1}^n{}^{n-1}{C}_{r-1}$

$[\because \frac{{}^n{C}_r}{{}^{n-2}{C}_{r-2}}=\frac{n(n-1)}{r(r-1)}]$

$=n(n-1)[{}^{n-2}{C}_0+{}^{n-2}{C}_1+\cdot \cdot \cdot +{}^{n-2}{C}_{n-2}]+n[{}^{n-1}{C}_0+{}^{n-1}{C}_1+\cdot \cdot \cdot +{}^{n-1}{C}_{n-1}]$

$=n\cdot (n-1)\cdot 2^{(n-2)}+n\cdot 2^{(n-1)}$

$=n\cdot 2^{(n-2)}\left(\right(n-1)+2)$

$=n(n+1)2^{n-2}$