Question

The value of $\left(\begin{array}{c}30\\ 0\end{array}\right)\left(\begin{array}{c}30\\ 10\end{array}\right)-\left(\begin{array}{c}30\\ 1\end{array}\right)\left(\begin{array}{c}30\\ 11\end{array}\right)+\left(\begin{array}{c}30\\ 2\end{array}\right)\left(\begin{array}{c}30\\ 12\end{array}\right)-\cdot \cdot \cdot \cdot \cdot \cdot +\left(\begin{array}{c}30\\ 20\end{array}\right)\left(\begin{array}{c}30\\ 30\end{array}\right)=$

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

Answer 1

Answer: (B)

${}^{30}{C}_{10}$

${}^{30}{C}_0{}^{30}{C}_{10}-{}^{30}{C}_1{}^{30}{C}_{11}+\cdots +{}^{30}{C}_{20}{}^{30}{C}_{30}$

This is the coefficient of $x^{20}$ in the expansion of $(1+x{)}^{30}(1-x{)}^{30}$

$\Rightarrow$ Coefficient of $x^{20}$ in the expansion of $(1-x^2{)}^{30}$

Now, Coefficient of $x^{20}$ in the expansion of $(1-x^2{)}^{30}$ is

$\sum _{r=0}^{30}(-1{)}^r{}^{30}{C}_r{x}^{2r}$

Substituting $r=10$, we get

Coefficient of $x^{20}$ as ${}^{30}{C}_{10}$

Hence, option B.