First slide

Binomial theorem for positive integral Index

Question

The coefficient of $x^{7}$ in the expression ${(1 + x)}^{10} + x {(1 + x)}^{9} + x^{2} {(1 + x)}^{8} + ...... + x^{10}$ is-

Moderate

A

330
B

210
C

420
D

120

Solution

Given series = $\frac{{(1 + x)}^{10} [1 - {(\frac{x}{1 + x})}^{11}]}{(1 - \frac{x}{1 + x})}$
$= \frac{{(1 + x)}^{10} [{(1 + x)}^{11} - x^{11}]}{{(1 + x)}^{11} \times \frac{1}{(1 + x)}}$
$= {(1 + x)}^{11} - x^{11}$
Coefficient of $x^{7}$ is $^{11} C_{7} =^{11} C_{4} = 330$

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