Question

The sum $\displaystyle \sum_{i-0}^{m} \binom{10}{i} \binom{20}{m-i},$ where $\binom{p}{q}=0 \, if \, p>q,$ is maximum when m is equal to

• A. 5
• B. 10
• C. 15
• D. 20

Answer 1

Answer: (C)

15

Answer 2

The correct answer is C:15
Given that;
$\displaystyle \sum^{m}_{i=0}(\frac{10}{i})(^{20}_{m-i})$
$=\displaystyle \sum^m_{i=0}10{c_{i}}\times20_{c_{m-i}}$
$=10_{c_{0}}\times20_{c_{m}}+10_{c_{1}}\times20_{c_{m-1}}+10_{c_{2}}\times20_{c_{m-2}}+.....10_{c_{m}}$
$=30_{c_{m}}$
We know that;
$\therefore m=\frac{n}{2}$ for $n_{c_{m}}$to be maximum
$\therefore m=\frac{30}{2}$
⇒$m=15$