Question

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

• A. $\left(\begin{array}{c}30\\ 15\end{array}\right)$
• B. $\left(\begin{array}{c}60\\ 15\end{array}\right)$
• C. $\left(\begin{array}{c}60\\ 30\end{array}\right)$
• D. $\left(\begin{array}{c}30\\ 10\end{array}\right)$

Answer 1

The correct option is D $\left(\begin{array}{c}30\\ 10\end{array}\right)$

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

WKT

$\Rightarrow (x+1{)}^{30}={}^{30}{C}_0+x\left({}^{30}{C}_1\right)+x^2\left({}^{30}{C}_2\right)+.........+x^{30}\left({}^{30}{C}_{30}\right)$

$\Rightarrow (x-1{)}^{30}=x^{30}\left({}^{30}{C}_0\right)-x^{29}\left({}^{30}{C}_1\right)+......{+}^{30}{C}_{30}$

Multiply the above $2$ equations $\xi$ comparing $x^{20}$ co-efficient.

$\Rightarrow (x+1{)}^{30}(x-1{)}^{30}={(x^2-1)}^{30}$

$\Rightarrow T_{10+1}{=}^{30}{C}_{10}{x}^{2\left(10\right)}\times 1^{20}={}^{30}{C}_{10}{x}^{20}$

$\Rightarrow {}^{30}{C}_{10}={}^{30}{C}_0{}^{30}{C}_{10}-{}^{30}{C}_1{}^{30}{C}_{11}+.......+{}^{30}{C}_{20}{}^{30}{C}_{30}$

$\therefore$ Required $={}^{30}{C}_{10}$

Hence, the answer is ${}^{30}{C}_{10}.$