题目

Problem

Find the coefficient of the term in $x^7$ in the binomial expansion of

\left(\frac38+4x\right)^{12}

(3)

解答

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在 $\left(a+b\right)^{12}$ 中，含 $x^7$ 的项一定来自 $(4x)^7$ ，所以另一部分是 $\left(\frac38\right)^5$ 。

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The term in $x^7$ is

\binom{12}{7} \left(\frac38\right)^5 (4x)^7.

So the coefficient of $x^7$ is

\begin{align*} \binom{12}{7} \left(\frac38\right)^5 4^7 =&\,792\left(\frac{243}{32768}\right)(16384)\\ =&\,96228. \end{align*}

Therefore, the coefficient is

\begin{align*} 96228. \end{align*}