The coefficient of $x^{7}$ in $(1 + 3 x - 2 x^{3})^{10}$ is equal to

62640

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26240

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64620

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Solution

The correct option is B $26240$
coefficient of $x^{7}$ in ${(1 + 3 x - 2 x^{3})}^{10}$
$x^{(n_{2} + 3 m_{3})}$ so $n_{2} + 3 n_{2} = 7$
Possible cases will be
$n_{2} = 7$ , $n_{3} = 0$ , $n_{1} = 3$
$n_{2} = 4$ , $n_{3} = 1$ , $n_{1} = 5$
$n_{2} = 1$ , $n_{3} = 2$ , $n_{1} = 7$
as $n_{3} < 3$
So coefficient of $x^{7}$ $= \frac{10 b}{3! 7!} 3^{7} + \frac{10 b}{4! 5!} 3^{4} (- 2) + \frac{10}{7! 2!} (3) (- 2)^{2}$
$= 26240$

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