Find the coefficient of
$x^{4}$ in the expansion of $(1 + 2 x + 3 x^{2})^{10}$ .

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Solution

General term of expansion is $\frac{10!}{a! b! c!} 2^{b} 3^{c} x^{b + 2 c}$ , where $a + b + c = 10$

We want to find the coefficient of $x^{4}$ , therefore $b + 2 c = 4$ . This is possible for,

$a = 8, b = 0, c = 2; a = 7, b = 2, c = 1; a = 6, b = 2, c = 0$

$∴$ the coefficient of $x^{4} = \frac{10!}{8! 0! 2!} 2^{0} 3^{2} + \frac{10!}{7! 2! 1!} 2^{2} 3^{1} + \frac{10!}{6! 4! 0!} 2^{4} 3^{0} = 405 + 4320 + 3360 = 8085$

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