Question

If $C_0,C_1,C_2,C_3,...,C_n$ denote the binomial coefficients in the expansion of ${(1+x)}^n$, then $1^2.C_1+2^2.C_2+3^2.C_3+...+n^2.C_n=$

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

Answer 1

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

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

${\sum }_{r=1}^n{r}^2{n_C}_r$

$={\sum }_{r=1}^n{r}^2\frac{n}{r}{{n-1}_C}_{r-1}$ using the formula ${n_C}_r=\frac{n}{r}\times {{n-1}_C}_{r-1}$

$=n{\sum }_{r=1}^{n}r\times {{n-1}_C}_{r-1}$

$=n{\sum }_{r=1}^n\{(r-1)+1\}{{n-1}_C}_{r-1}$

$=n{\sum }_{r=1}^n(r-1){{n-1}_C}_{r-1}+n{\sum }_{r=1}^n{{n-1}_C}_{r-1}$

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

$=n(n-1)(0+{{n-2}_C}_0+{{n-2}_C}_1+{{n-2}_C}_2+...+{{n-2}_C}_{n-2})+n({{n-1}_C}_0+{{n-1}_C}_1+{{n-1}_C}_2+...+{{n-1}_C}_{n-1})$

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

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