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Question

Coefficients of $x to the power of 18 end exponent$ in $open parentheses 1 plus x plus 2 x to the power of 2 end exponent plus 3 x to the power of 3 end exponent plus horizontal ellipsis plus 18 x to the power of 18 end exponent close parentheses to the power of 2 end exponent$ is equal to

995
1005
1235
None of these

The correct answer is: 1005

Coefficient of $x to the power of 18 end exponent$ in $open parentheses 1 plus x plus 2 x to the power of 2 end exponent plus 3 x to the power of 3 end exponent plus horizontal ellipsis plus 18 x to the power of 18 end exponent close parentheses to the power of 2 end exponent$
=Coefficient of $x to the power of 18 end exponent$ in $open parentheses 1 plus x plus 2 x to the power of 2 end exponent plus 3 x to the power of 3 end exponent plus horizontal ellipsis plus 18 x to the power of 18 end exponent close parentheses cross times open parentheses 1 plus x plus 2 x to the power of 2 end exponent plus 3 x to the power of 3 end exponent plus horizontal ellipsis plus 18 x to the power of 18 end exponent close parentheses$
$equals 1 cross times 18 plus 1 cross times 17 plus 2 cross times 16 plus horizontal ellipsis plus 17 cross times 1 plus 18 cross times 1$
$equals 36 plus not stretchy sum from r equals 1 to 17 of r open parentheses 18 minus r close parentheses$
$equals 36 plus 18 not stretchy sum from r equals 1 to 17 of r minus not stretchy sum from r equals 1 to 17 of r to the power of 2 end exponent$
$equals 1005$

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