Find integer solutions to a three-variable equation Consider a three-variable equation \begin{equation}
\overline{x y z}=\frac{3}{2} x ! y ! z !
\end{equation}
What I have tried:
\begin{equation}
\overline{x y z}=100 x+10 y+z
\end{equation}
\begin{equation}
\begin{array}{l}
\max (\overline{x y z})=999 \Rightarrow \max (x ! y ! z !)=666 \Rightarrow \\
\Rightarrow \operatorname{max}(\overline{x y z})=555 \text {  } \\
\therefore \operatorname{max}(\overline{x y z})=555 \\
\therefore \operatorname{max}(x ! y ! z !)=370
\end{array}
\end{equation}
So as you see I found max possible values, unfortunately, I don't have ideas on how to solve this correctly and how to minimize the set of possible solutions...
 A: We can assume that $x\geq y \geq z$.  You have correctly deduced that $x\leq 5$.  If $z=1$, then the left side is odd.  But every factorial bigger than $1!$ is even, so the only possibility is $211$, which doesn't work.  So $z\geq 2$.
Could $x=5$?  The smallest possible number would be $522$ but that already makes the right side too big, so "no".  We have $x\leq 4$.
That leaves only $444, 443, 442, 433, 432$ to check. And $432$ works.
Edit:  Actually, we can eliminate $443$ and $433$ because they're odd.  We can also eliminate non-multiples of $3$, so that leaves only $444$ and $432$.  Since $444$ is a multiple of $37$, it won't work either.  (Which I know because $111=3\cdot 37.$
A: A slightly different method that doesn't require the assumption $x \ge y \ge z$:
First, we note that no digit can be larger than $5$, and that $N = \frac32 x!y!z!$ must be even.
If no digits are greater than $2$, then  $x!y!z! \le 8$. Hence at least one digit must be $3$ or greater. Then $3 \mid x!y!z! \implies 9 \mid N$.
We can't make $18$ without at least one digit $6$ or greater, so for any solution, $x+y+z=9$. That narrows our possibilities by a huge amount, to just
$$144,234,252,324,342,414,432,450,504,522,540$$
Now each of those contains enough $2$s in factorials that we can eliminate anything not divisible by $16$. That leaves only $144$ and $432$. $144$ doesn't work, and B Goddard already noted that $432$ does.
