Question

The value of
$\left(\genfrac{}{}{0ex}{}{30}{0}\right)\left(\genfrac{}{}{0ex}{}{30}{10}\right)-\left(\genfrac{}{}{0ex}{}{30}{1}\right)\left(\genfrac{}{}{0ex}{}{30}{11}\right)+\left(\genfrac{}{}{0ex}{}{30}{2}\right)\left(\genfrac{}{}{0ex}{}{30}{12}\right)+....+\left(\genfrac{}{}{0ex}{}{30}{20}\right)\left(\genfrac{}{}{0ex}{}{30}{30}\right)=$

• A. ${}^{60}{C}_{20}$
• B. ${}^{30}{C}_{10}$
• C. ${}^{60}{C}_{30}$
• D. ${}^{40}{C}_{30}$

Answer 1

Answer: (B)

$(1-x{)}^{30}={}^{30}{C}_0{x}^0-{}^{30}{C}_1{x}^1+{}^{30}{C}_2{x}^2+\dots .+(-1{)}^{30}{}^{30}{C}_{30}{x}^{30}...(i)$
$(x+1{)}^{30}={}^{30}{C}_0{x}^{30}+{}^{30}{C}_1{x}^{29}+{}^{30}{C}_2{x}^{28}+\dots .+{}^{30}{C}_{10}{x}^{20}+\dots .+{}^{30}{C}_{30}{x}^0...(ii)$
Multiplying (i) and (ii) and equating the coefficient of $x^{20}$ on both sides, we get required sum is equal to coefficient of $x^{20}$ in ${\left(1-x^2\right)}^{30}$, which is given by ${}^{30}{C}_{10}$.