Question

The value of $\frac{1}{2}{}^n{C}_1.{}^{n-1}{C}_1+\frac{2}{3}{}^n{C}_2+\frac{3}{4}{}^n{C}_c.{}^{n-1}{C}_3+\dots \dots$ is

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

Answer 1

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

Given sum of the series
$={\sum }_{r=1}^{n-1}{\frac{r}{r+1}}^n{C}_r{}^{n-1}{C}_r$
$={\sum }_{r=1}^{n-1}{}^{n-1}{C}_{r-1}{}^n{C}_{r+1}$
$(1+x{)}^{n-1}{=}^{n-1}{C}_0{+}^{n-1}{C}_x{+}^{n-1}{C}_2{x}^2+\dots \dots$
$(1+x{)}^n{=}^n{C}_0{x}^n{+}^n{C}_1{x}^{n-1}{+}^n{C}_2{x}^{n-2}+\dots \dots$
$\therefore {}^{n-1}{C}_0{}^n{C}_1+{}^{n-1}{C}_1+{}^n{C}_3+\dots \dots$
= coefficient of $x^{n-2}in(1+x{)}^{n-1}(1+x){}^n{}^{2n-1}{C}_{n-2}$