First slide

Binomial theorem for positive integral Index

Question

The sum of the series $\sum_{r = 0}^{15} {}^{30}C_{r}$

Moderate

A

$2^{29} + \frac{1}{2} {}^{30}C_{15}$
B

$2^{30} + \frac{1}{2} {}^{30}C_{15}$
C

$2^{29} + {}^{30}C_{15}$
D

$\frac{2^{29} + {}^{30}C_{15}}{2}$

Solution

Let $S = \sum_{r = 0}^{15} {}^{30}C_{r} = {}^{30}C_{0} + {}^{30}C_{1} + {}^{30}C_{2} + .... + {}^{30}C_{15}$

Also $S = {}^{30}C_{30} + {}^{30}C_{29} + {}^{30}C_{28} + .... + {}^{30}C_{15} (∵ {}^{n}C_{r} = {}^{n}C_{n - r})$
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By adding $2 S = {}^{30}C_{0} + {}^{30}C_{1} + .... + {}^{30}C_{30} + {}^{30}C_{15}$

$2 S = 2^{30} + {}^{30}C_{15}$
$\Rightarrow S = 2^{29} + \frac{{}^{30}C_{15}}{2}$

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