First slide

Binomial theorem for positive integral Index

Question

The coefficient of x⁷ in the expression $(1 + x)^{10} + x (1 + x)^{9} + x^{2} (1 + x)^{8} + \dots + x^{10}$ is

Easy

A

420
B

330
C

210
D

120

Solution

Given expression $(1 + x)^{10} + x (1 + x)^{9} + x^{2} (1 + x)^{8} + \dots + x^{10}$
The above expression is geometric series having first term, $a = (1 + x)^{10}$
Common ratio, $r = \frac{x}{1 + x}$ and number of terms, n = 11
sum of geometric series $= \frac{a (1 - r^{n})}{1 - r} = \frac{(1 + x)^{10} (1 - {(\frac{x}{1 + x})}^{11})}{1 - \frac{x}{1 + x}}$
$\begin{matrix} = \frac{(1 + x)^{10} (\frac{(1 + x)^{11} - x^{11}}{(1 + x)^{11}})}{\frac{1 + x - x}{1 + x}} \\ = (1 + x)^{11} - x^{11} \end{matrix}$
The coefficient of x7 an the given expression
= coefficient of x7 in the expansion of $[(1 + x)^{11} - x^{11}]$
= coefficient of x7 in the expansion of $(1 + x)^{11}$
$\begin{matrix} =^{11} C_{7} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2} \\ = 11 \times 10 \times 3 = 330 \end{matrix}$

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