Question

${}^{15}{C}_2+{2.}^{15}{C}_3+{3.}^{15}{C}_4+\dots ..+{14.}^{15}{C}_{15}=$

• A. ${14.2}^{15}+1$
• B. ${13.2}^{14}+1$
• C. ${14.2}^{15}-1$
• D. ${13.2}^{14}-1$

Answer 1

The correct option is B ${13.2}^{14}+1$

$(1+x{)}^{15}=1+x{}^{15}{C}_1+x^2{}^{15}{C}_2+...+x^{15}{}^{15}{C}_{15}$
Dividing both the sides with $x$.
$\frac{(1+x{)}^{15}}{x}=\frac{1}{x}+{}^{15}{C}_1+x{}^{15}{C}_2+...+x^{14}{}^{15}{C}_{15}$
Differentiating both the sides with respect to $x$
$\frac{15x(1+x{)}^{14}-(1+x{)}^{15}}{x^2}=\frac{-1}{x^2}+{}^{15}{C}_2+...+14x^{13}{}^{15}{C}_{15}$ ...(i)
Substituting $x=1$ in equation (i), we get
$15(2{)}^{14}-(2{)}^{15}=-1+{}^{15}{C}_2+...+14{}^{15}{C}_{15}$
$2^{14}(15-2)+1={}^{15}{C}_2+...+14{}^{15}{C}_{15}$
$2^{14}\left(13\right)+1={}^{15}{C}_2+...+14{}^{15}{C}_{15}$
Hence answer is Option B