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. $\left(\begin{array}{c}30\\ 10\end{array}\right)$
• B. $\left(\begin{array}{c}30\\ 15\end{array}\right)$
• C. $\left(\begin{array}{c}60\\ 30\end{array}\right)$
• D. $\left(\begin{array}{c}31\\ 10\end{array}\right)$

Answer 1

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

To find the value of
$(\frac{30}{0})(\frac{30}{10})-(\frac{30}{1})(\frac{30}{11})+(\frac{30}{2})(\frac{30}{12})+......(\frac{30}{20})(\frac{30}{30})$
The above expression is the coefficient of $x^{20}$ in the Product
${(1+x)}^{30}{(1-x)}^{30}={(1-x^2)}^{30}$
The given expression is
$=(\frac{30}{10})$
So, Option A is correct