First slide

Binomial theorem for positive integral Index

Question

Coefficient of the term independent of x in the expan-sion of $(1 + x)^{2 n} {(\frac{x}{1 - x})}^{- 2 n}$ is :

Moderate

A

$^{2 n} C_{n}$
B

${(- 1)^{n}}^{2 n} C_{n}$
C

$^{n^{2} - n} C_{n}$
D

0

Solution

Coefficient of the term independent of x in the expan-sion of
$(1 + x)^{2 n} {(\frac{x}{1 - x})}^{- 2 n}$
or $\frac{(1 + x)^{2 n} (1 - x)^{2 n}}{x^{2 n}}$
= coefficient of $x^{2 n} in {(1 - x^{2})}^{2 n}$
= coefficient of $t^{n} in (1 - t)^{2 n}$
$=^{2 n} C_{n} (- 1)^{n}$

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