First slide

Binomial theorem for positive integral Index

Question

The coefficient of $x^{9}$ in the expansion of $E$ $= (1 + x)^{9^{}} + (1 + x)^{10} + \dots + (1 + x)^{100}$ is

Moderate

A

$^{101} C_{9}$
B

$^{101} C_{10}$
C

$^{100} C_{9}$
D

$^{100} C_{10}$

Solution

$\begin{aligned} E = \frac{(1 + x)^{9} [(1 + x)^{92} - 1]}{1 + x - 1} \\ = \frac{(1 + x)^{101} - (1 + x)^{9}}{x} \end{aligned}$
Coefficient of $x^{9}$ in $E$
$=$ coefficient of $x^{10}$ in $[(1 + x)^{101} - (1 + x)^{9}]$
$=^{101} C_{10}$

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