Coefficient of $x^{18} i n (1 + x + 2 x^{2} + 3 x^{3} + . . . . . . . + 18 x^{18}$ is equal to

995

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1005

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1235

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none of these

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Solution

The correct option is B
1005

Coefficient of $x^{18} i n (1 + x + 2 x^{2} + 3 x^{3} + . . . . . . . + 18 x^{18})^{2}$

$=$ Coefficient of $x^{18} i n (1 + x + 2 x^{2} + 3 x^{3} + . . . . + 18 x^{18})$

$\times (1 + x + 2 x^{2} + 3 x^{3} + . . . . + 18 x 18)$

$= 1 \times 18 + 1 \times 17 + 2 \times 16 + . . . . . + 17 \times 1 + 18 \times 1$

$= 36 + (1 \times 17 + 2 \times 16 + . . . . . + 17 \times 1$

$= 36 + 17 \sum r = 1 r - 17 \sum r = 1 r^{2}$

$= 36 + 18 \frac{18 (17)}{2} - \frac{17 (18) (35)}{6}$

$= 1005$

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