First slide

Binomial theorem for positive integral Index

Question

$Sum of the last 20 coefficients in the expansion of (1 + x)^{39}, when expanded$ $in ascending powers of x, is$

Moderate

A

$2^{19}$
B

$2^{18}$
C

$^{40} C_{20} - 2^{19}$
D

$2^{38}$

Solution

$There are 40 terms in the expansion of (1 + x)^{39} :Sum of last 20$ $coefficients is$
$\begin{array}{l} S =^{39} C_{20} +^{39} C_{21} + \dots +^{39} C_{38} +^{39} C_{39} \\ \Rightarrow S =^{39} C_{19} +^{39} C_{18} + \dots +^{39} C_{1} +^{39} C_{0} \\ Adding the above two expansions we get 2 S =^{39} C_{0} +^{39} C_{1} + \dots +^{39} C_{39} \\ = 2^{39} \Rightarrow S = 2^{38} . \end{array}$

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