First slide

Binomial theorem for positive integral Index

Question

Sum of the last 30 coefficients of powers of $x$ in the binomial expansion of $(1 + x)^{59}$ is

Moderate

A

$2^{29}$
B

$2^{28}$
C

$2^{59} - 2^{29}$
D

$2^{58}$

Solution

Let
$S =$ some of the last 30 coefficients in $(1 + x)^{59}$
$=^{59} C_{30} +^{59} C_{31} + \dots +^{59} C_{59}$ (1)
Using $^{n} C_{r} =^{n} C_{n - r}$ , we get
$S =^{59} C_{29} +^{59} C_{28} + \dots +^{59} C_{0}$ (2)
Adding (1) and (2), we get
$\begin{array}{l} 2 S =^{59} C_{0} +^{59} C_{1} + \dots +^{59} C_{59} = 2^{59} \\ \Rightarrow S = 2^{58} \end{array}$

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