Binomial Theorem Ques 5
- The sum $\sum_{i=0}^m\binom{10}{i}\binom{20}{m-i}$, where $\binom{p}{q}=0$ if $p>q$, is maximum when $m$ is equal to
(2002, 1M)
(a) $5$
(b) $10$
(c) $15$
(d) $20$
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Answer:
Correct Answer: 5.(c)
Solution: (c) $\sum_{i=0}^m\binom{10}{i}\binom{20}{m-i}$ is the coefficient of $x^m$ in the expansion of $(1+x)^{10}(x+1)^{20}$.
$\Rightarrow \quad \sum_{i=0}^m\binom{10}{i}\binom{20}{m-i}$ is the coefficient of $x^m$ in the expansion of $(1+x)^{30}$
i.e. $\quad \sum_{i=0}^m\binom{10}{i}\binom{20}{m-i}={ }^{30} C_m=\binom{30}{m}$
and we know that, $\binom{n}{r}$ is maximum, when
$ \binom{n}{r}_{\max }= \begin{cases}r=\frac{n}{2}, & \text { if } n \in \text { even. } \\ r=\frac{n \pm 1}{2}, & \text { if } n \in \text { odd.}\end{cases} $
Hence, $\binom{30}{m}$ is maximum when $m=15$.
BETA
