Question

The value of the sum of the series ${}^{14}{C}_0{.}^{15}{C}_1{+}^{14}{C}_1{.}^{15}{C}_2{+}^{14}{C}_2{.}^{15}{C}_3+...{+}^{14}{C}_{14}{.}^{15}{C}_{15}$ is

• A. ${}^{29}{C}_{12}$
• B. ${}^{29}{C}_{10}$
• C. ${}^{29}{C}_{14}$
• D. ${}^{29}{C}_{16}$

Answer 1

The correct option is C ${}^{29}{C}_{14}$

${(1+x)}^{14}={{}^{14}C}_0+{{}^{14}C}_{1}x+{{}^{14}C}_2{x}^2+...+{{}^{14}C}_{14}{x}^{14}$ ....(1)

and ${(x+1)}^{15}={{}^{15}C}_0{x}^{15}+{{}^{15}C}_1{x}^{14}+{{}^{15}C}_2{x}^{13}+...+{{}^{15}C}_{15}$

Multiplying (1) and (2) and equating the coefficients of $x^{14}$ we get

${{}^{14}C}_0.{{}^{15}C}_1+{{}^{14}C}_1.{{}^{15}C}_2+{{}^{14}C}_2.{{}^{15}C}_3+...+{{}^{14}C}_{14}.{{}^{15}C}_{15}$

$=$ the coefficients of $x^{14}$ in ${(1+x)}^{29}={{}^{29}C}_{14}$.