What are the limits between which the value of the following expression can vary if we parenthesize it in every possible way? $1: 2: 3: 4: 5: 6: 7: 8: 9$.
The expression can be written as a single fraction in a given parentheses configuration, where each of the numbers $1, 2, \ldots, 9$ appears exactly once as a factor in the numerator or the denominator. The division sign between 1 and 2 is the first, so in every possible parentheses configuration, 1 is in the numerator and 2 is in the denominator.

Accordingly, the numerator is at least 1 and at most $1 \cdot 3 \cdot 4 \cdot \ldots \cdot 9$, while the denominator is at least 2 and at most $2 \cdot 3 \cdot \ldots \cdot 9$. Thus, the value of the fraction is at least $\frac{1}{2 \cdot 3 \cdot \ldots \cdot 9}=\frac{1}{362800}$, and its maximum value is at most $\frac{1 \cdot 3 \cdot \ldots \cdot 9}{2}=90720$. These values are indeed taken by the expression in the following parentheses configurations:

$$
\begin{aligned}
& (((((((1: 2): 3): 4): 5): 6): 7): 8): 9=\frac{1}{362880}, \text { and } \\
& \quad 1:(((((((2: 3): 4): 5): 6): 7): 8):) 9=90720
\end{aligned}
$$

Therefore, the value of the expression varies between $\frac{1}{362880}$ and 90720.