The coefficient of $x^{8}$ in the polynomial $(x - 1) (x - 2) (x - 3) . . . (x - 10)$ is

1025

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1240

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1320

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1440

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Solution

The correct option is D $1320$
$(x - 1) (x - 2) \Rightarrow x^{2} - 3 x + 2 \Rightarrow x^{2} - (2 + 1) x + (2 + 1)$
$(x - 1) (x - 2) (x - 3) \Rightarrow x^{3} - (1 + 2 + 3) x^{2} + (1 \times 2 + 1 \times 3 + 2 \times 3) x + (1 \times 2 \times 3)$
$(x - a_{1}) (x - a_{2}) . . . . (x - a_{n}) = x^{n} - (a_{1} + a_{2} + . . . a_{n}) x^{n - 1} + (a_{1} a_{2} . . . . a_{n})$
$(x - 1) (x - 2) (x - 3) . . . . (x - 10)$
$x^{8} \Rightarrow [2 + 3 + 4 + . . . + 10] + 2 [3 + 4 + . . . . + 10] + . . .9 \times 10$
$= 1 \times 54 + 2 \times 52 + 3 \times 49 + . . . .9 \times 10$
$= 1320 \Rightarrow$ Required coefficient of $x^{8}$

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