18.5.7 $\star$ Find all positive integer solutions to the equation $n!-x!+y!+z!$.

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18.5.7 $\star$ Find all positive integer solutions to the equation $n! - x! + y! + z! = 0$.

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Suppose $x \leqslant y \leqslant z$. Clearly, $n \geqslant z+1$. Therefore,
$$
(z+1)!\leqslant n!=x!+y!+z!\leqslant 3 \cdot z!,
$$

thus $z \leqslant 2$. By calculation, we know that $x=y=z=2, n=3$ is the only solution to the original equation.