First slide

Binomial theorem for positive integral Index

Question

The sum of the series $S =^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + \dots +^{20} C_{10}$ is

Moderate

A

$- (^{20} C_{10})$
B

$\frac{1}{2} (^{20} C_{10})$
C

0
D

$^{20} C_{10}$

Solution

Using $^{n} C_{r} =^{n} C_{n - r}$ , we can write
$\begin{array}{l} 2 S =^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + \dots^{20} C_{10} +^{20} C_{20} -^{20} C_{19} + \\ ^{20} C_{18} - \dots +^{20} C_{10} \\ \Rightarrow S = 0 + \frac{1}{2} (^{20} C_{10}) = \frac{1}{2} (^{20} C_{10}) \end{array}$

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