Question

The value of
$\left(\genfrac{}{}{0ex}{}{30}{0}\right)\left(\genfrac{}{}{0ex}{}{30}{10}\right)-\left(\genfrac{}{}{0ex}{}{30}{1}\right)\left(\genfrac{}{}{0ex}{}{30}{11}\right)+\left(\genfrac{}{}{0ex}{}{30}{2}\right)\left(\genfrac{}{}{0ex}{}{30}{12}\right).....+\left(\genfrac{}{}{0ex}{}{30}{20}\right)\left(\genfrac{}{}{0ex}{}{30}{30}\right)$
is where $\left(\genfrac{}{}{0ex}{}{n}{r}\right){=}^n{C}_r$

• A. $\left(\genfrac{}{}{0ex}{}{30}{10}\right)$
• B. $\left(\genfrac{}{}{0ex}{}{30}{15}\right)$
• C. $\left(\genfrac{}{}{0ex}{}{60}{30}\right)$
• D. $\left(\genfrac{}{}{0ex}{}{31}{10}\right)$

Answer 1

Answer: (A)

To find
${}^{30}{C}_0^{30}{C}_{10}{-}^{30}{C}_1^{30}{C}_{11}{+}^{30}{C}_2^{30}{C}_{12}-....{+}^{30}{C}_{20}^{30}{C}_{30}$
We know that
${\left(1+x\right)}^{30}{=}^{30}{C}_0{+}^{30}{C}_{1}x{+}^{30}{C}_2{x}^2$
$+....{+}^{30}{C}_{20}{x}^{20}+...{.}^{30}{C}_{30}{x}^{30}....\left(1\right)$
${\left(x-1\right)}^{30}{=}^{30}{C}_0{x}^{30}{-}^{30}{C}_1{x}^{29}+....{+}^{30}{C}_{10}{x}^{20}$
${-}^{30}{C}_{11}{x}^{19}{+}^{30}{C}_{12}{x}^{18}+...{.}^{30}{C}_{30}{x}^0....\left(2\right)$
Multiplying $e{q}^n\left(1\right)$ and $\left(2\right)$ and equating the coefficients of $x^{20}$ on both sides, we get
${}^{30}{C}_{10}{=}^{30}{C}_0^{30}{C}_{10}{-}^{30}{C}_1^{30}{C}_{11}{+}^{30}{C}_2^{30}{C}_{12}-....{+}^{30}{C}_{20}^{30}{C}_{30}$
$\therefore$ Req. value is ${}^{30}{C}_{10}$