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Maths-

General

Easy

Question

In the expansion of $left parenthesis 1 plus x plus x to the power of 2 end exponent plus x to the power of 3 end exponent right parenthesis to the power of 6 end exponent comma$ then coefficient of $x to the power of 14 end exponent$ is

130
120
128
125

The correct answer is: 120

$left parenthesis 1 plus x plus x to the power of 2 end exponent plus x to the power of 3 end exponent right parenthesis to the power of 6 end exponent equals open parentheses 1 plus x close parentheses to the power of 6 end exponent open parentheses 1 plus x to the power of 2 end exponent close parentheses to the power of 6 end exponent$
$equals open parentheses blank to the power of 6 end exponent C subscript 0 end subscript plus blank to the power of 6 end exponent C subscript 1 end subscript x plus blank to the power of 6 end exponent C subscript 2 end subscript x to the power of 2 end exponent plus blank to the power of 6 end exponent C subscript 3 end subscript x to the power of 3 end exponent plus blank to the power of 6 end exponent C subscript 4 end subscript x to the power of 4 end exponent plus blank to the power of 6 end exponent C subscript 5 end subscript x to the power of 5 end exponent plus blank to the power of 6 end exponent C subscript 6 end subscript x to the power of 6 end exponent close parentheses cross times left parenthesis blank to the power of 6 end exponent C subscript 0 end subscript plus blank to the power of 6 end exponent C subscript 1 end subscript x to the power of 2 end exponent plus blank to the power of 6 end exponent C subscript 2 end subscript x to the power of 4 end exponent plus$
$blank to the power of 6 end exponent C subscript 3 end subscript x to the power of 6 end exponent plus blank to the power of 6 end exponent C subscript 4 end subscript x to the power of 8 end exponent plus blank to the power of 6 end exponent C subscript 5 end subscript x to the power of 10 end exponent plus blank to the power of 6 end exponent C subscript 6 end subscript x to the power of 12 end exponent right parenthesis$
$therefore$ Coefficient of $x to the power of 14 end exponent$ in $left parenthesis 1 plus x plus x to the power of 2 end exponent plus x to the power of 3 end exponent right parenthesis to the power of 6 end exponent$
$equals blank to the power of 6 end exponent C subscript 2 end subscript bullet blank to the power of 6 end exponent C subscript 6 end subscript plus blank to the power of 6 end exponent C subscript 4 end subscript bullet blank to the power of 6 end exponent C subscript 5 end subscript plus blank to the power of 6 end exponent C subscript 6 end subscript bullet blank to the power of 6 end exponent C subscript 4 end subscript$
$equals 15 plus 90 plus 15 equals 120$

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