The coefficient of $x^{30}$ in the expansion of $(1 + 2 x + 3 x^{2} + . . . . .21 x^{20})^{2}$ is

2706

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2450

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1481

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256

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Solution

The correct option is B $2706$
$(1 + 2 x + 3 x^{2} + . . . .21 x^{20}) (21 x^{20} + 20 x^{19} + 19 x^{18} + . . . . .2 x + 1)$
coeff. of $x^{30}$ is $11.21 + 12.20 + . . . . . . + 21.11$
$= 2 [11.21 + 12.20 + 13.19 + 14.18 + 15.17] + 16.16$
$= 2 [231 + 240 + 247 + 252 + 255] + 256$
$= 2 [1225] + 256$
$= 2450 + 256$
$= 2706$

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