Question

Assertion :The sum of $\sum _{i=0}^m\left(\begin{array}{c}10\\ i\end{array}\right)\left(\begin{array}{c}20\\ m-i\end{array}\right)$ will be maximum if $m=15$ Reason: The expression ${}^m{C}_r$ attain maximum value if $r=\frac{m}{2}.$

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A),
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A)is true but (R) is false.
• D. (A)is false but (R) is true.

Answer 1

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A),

$S=\sum _{r=0}^m\left(\begin{array}{c}10\\ i\end{array}\right)\left(\begin{array}{c}20\\ m-i\end{array}\right)=\left(\begin{array}{c}10\\ 0\end{array}\right)\left(\begin{array}{c}20\\ m\end{array}\right)+\left(\begin{array}{c}10\\ 1\end{array}\right)\left(\begin{array}{c}20\\ m-1\end{array}\right)+.....+\left(\begin{array}{c}10\\ m\end{array}\right)\left(\begin{array}{c}20\\ 0\end{array}\right)$
$=$ Coefficient of $x^m$ in the product of $(1+x{)}^{10}(1+x{)}^{20}{=}^{30}{C}_m$
Now ${}^{30}{C}_m$ will be maximum if $m=\frac{30}{2}=15$

Hence, both assertion and reason are correct. And assertion is followedby reason.