By AM-GM,
\begin{align*}
a + b + c + d &= a + \frac{b}{2} + \frac{b}{2} + \frac{c}{3} + \frac{c}{3} + \frac{c}{3} + \frac{d}{4} + \frac{d}{4} + \frac{d}{4} + \frac{d}{4} \\
&\ge 10 \sqrt[10]{a \left( \frac{b}{2} \right)^2 \left( \frac{c}{3} \right)^3 \left( \frac{d}{4} \right)^4} \\
&= 10 \sqrt[10]{\frac{ab^2 c^3 d^4}{27648}}.
\end{align*}Since $a + b + c + d = 10,$
\[ab^2 c^3 d^4 \le 27648.\]Equality occurs when $a = 1,$ $b = 2,$ $c = 3,$ and $d = 4,$ so the maximum value is $\boxed{27648}.$
Question:
Let $a,$ $b,$ $c,$ and $d$ be positive real numbers such that $a + b + c + d = 10.$  Find the maximum value of $ab^2 c^3 d^4.$
Answer:
Let's think step by step.

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27648
Let $a,$ $b,$ $c,$ and $d$ be positive real numbers such that $a + b + c + d = 10.$  Find the maximum value of $ab^2 c^3 d^4.$
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