Question

The sum $\sum _{i=0}^m\left(\begin{array}{c}10\\ i\end{array}\right)\left(\begin{array}{c}20\\ m-i\end{array}\right)$ for $i\in [0,m]$, where $(p,q)=0,$ if $p>q$ is maximum when $\text{'}m\text{'}$ is

• A. $5$
• B. $10$
• C. $15$
• D. $20$

Answer 1

The correct option is C

$15$

Explanation for the correct option:

Given: $\sum _{i=0}^m\left(\begin{array}{c}10\\ i\end{array}\right)\left(\begin{array}{c}20\\ m-i\end{array}\right)$

$\sum _{i=0}^m\left(\begin{array}{c}10\\ i\end{array}\right)\left(\begin{array}{c}20\\ m-i\end{array}\right)$ is the coefficient of ${\left(x\right)}^m$ in the binomial expansion of ${\left(1+x\right)}^{10}{\left(1+x\right)}^{20}$

$\sum _{i=0}^m\left(\begin{array}{c}10\\ i\end{array}\right)\left(\begin{array}{c}20\\ m-i\end{array}\right)$ is the coefficient of ${\left(x\right)}^m$ in the ${}^{30}{C}_m=\left(\begin{array}{c}30\\ m\end{array}\right)$

We know that $\left(\begin{array}{c}n\\ r\end{array}\right)$ is maximum when $\left(\begin{array}{c}n\\ r\end{array}\right)=\left\{\begin{array}{l}r=\frac{n}{2},n\in Even\\ r=\frac{n+1}{2},n\in Odd\end{array}\right.$

$\left(\begin{array}{c}30\\ m\end{array}\right)$ is maximum when $m=\frac{30}{2}=15$

Hence option(C) i.e. $15$ is correct.