Question

${}^{15}{C}_0{.}^5{C}_5{+}^{15}{C}_1{.}^5{C}_4{+}^{15}{C}_2{.}^5{C}_3{+}^{15}{C}_3{.}^5{C}_2$ ${+}^{15}{C}_4{.}^5{C}_1$ is equal to

• A. $2^{20}-2^5$
• B. $\frac{20!}{5!15!}$
• C. $\frac{20!}{5!15!}-1$
• D. $\frac{20!}{5!15!}-\frac{15!}{5!10!}$
• E. $\frac{15!}{5!10!}$

Answer 1

Answer: (D)

Now, coefficient of $x^{15}$ in ${(1+x)}^{20}<br>$= coefficient of $x^{15}$ in ${(1+x)}^{15}{(1+x)}^5$
$\Rightarrow$ ${}^{20}{C}_{15}=$ coefficient of $x^{15}$ in ${(}^{15}{C}_0{x}^{15}{+}^{15}{C}_1{x}^{14}$ ${+}^{15}{C}_2{x}^{13}{+}^{15}{C}_3{x}^{12}{+}^{15}{C}_4{x}^{11}{+}^{15}{C}_5{x}^{10})$ ${(}^5{C}_0{x}^5{+}^5{C}_1{x}^4{+}^5{C}_2{x}^3{+}^5{C}_3{x}^2$ ${+}^5{C}_{4}x{+}^5{C}_5)$
$\Rightarrow$ ${}^{20}{C}_{15}{=}^{15}{C}_0{.}^5{C}_5{+}^{15}{C}_1{.}^5{C}_4{+}^{15}{C}_2{.}^5{C}_3$ ${+}^{15}{C}_3{.}^5{C}_2{+}^{15}{C}_4{.}^5{C}_1{+}^{15}{C_5}^5{C}_0$
$\Rightarrow$ ${}^{15}{C}_0{.}^5{C}_5{+}^{15}{C}_1{.}^5{C}_4{+}^{15}{C}_2{.}^5{C}_3{+}^{15}{C}_3{.}^5{C}_2$ ${+}^{15}{C}_4{.}^5{C}_1{=}^{20}{C}_{15}{-}^{15}{C_5}^5{C}_0$
$=\frac{20!}{5!5!}-\frac{15!}{5!10!}$