A country has $k$ provinces, each with $a_{1}, a_{2}, \cdots, a_{k}$ cities, respectively. It is planned to establish some (two-way) air routes between these cities so that from any city, one can reach any other city through one or more air routes. If no air routes are set up between cities within the same province, and it is required that the total number of air routes be as few as possible, how many different air route design methods are there?
Let $n=\sum_{i=1}^{k} a_{i}$, and let $S=\{1,2, \cdots, n\}$ represent all cities, where
$$
\begin{array}{l}
A_{1}=\left\{1,2, \cdots, a_{1}\right\}, \\
A_{2}=\left\{a_{1}+1, a_{1}+2, \cdots, a_{1}+a_{2}\right\}, \\
\cdots \cdots \\
A_{k}=\left\{\sum_{i=1}^{n-1} a_{i}+1, \cdots, \sum_{i=1}^{k} a_{i}\right\}
\end{array}
$$

represent the cities of the $1,2, \cdots, k$ provinces, respectively.
It is easy to see that the minimum number of flight routes must be $n-1$. For any set of flight routes that meet the requirements, each city must have at least one flight route, and at least two cities must have exactly one flight route. Let the smallest number among these cities be $b$, and assume $b \in A_{i}$, and $b$ is connected to city $c$ by a flight route. Construct a segmented sequence with $k+1$ segments, where the first $1,2, \cdots, k$ segments are of lengths
$a_{1}-1, a_{2}-1, \cdots, a_{k}-1$, and the $(k+1)$-th segment is of length $k-2$, all composed of elements from $S$, with a total length of $n-2$.
If $\left|A_{i}\right|>1$, then place $c$ as the first element of the $i$-th segment;
If $\left|A_{i}\right|=1$, then place $c$ as the first element of the $(k+1)$-th segment.

After removing $b$, the remaining cities can still be connected by flight routes, and the number of flight routes remains minimal. Therefore, the previous operation can be repeated, removing one city each time and recording the city connected to it by a unique flight route in the corresponding segment of the segmented sequence.

After $n-2$ operations, only two cities remain, and all elements of the segmented sequence are filled. This is because there must be $k-2$ provinces whose cities are all removed, and the remaining two provinces each have one city left. If all cities of the $i$-th province are removed, then the first $a_{i}-1$ cities removed are recorded as the elements of the $i$-th segment, and the last city removed is recorded as an element of the $(k+1)$-th segment. For provinces with only one city left, the $(k+1)$-th segment does not contain the corresponding element, so the first $k$ segments are all filled, and the $(k+1)$-th segment contains $k-2$ elements, also filling it completely.

Thus, each flight route design that meets the requirements can correspond to a sequence, where the elements of the $i$-th segment ($1 \leqslant i \leqslant k$) can be any element of $S-A_{i}$, and the $(k+1)$-th segment can be any element of $S$. Therefore, the number of different sequences is
$$
\begin{array}{l}
\prod_{i=1}^{k}\left|S-A_{i}\right||S|^{k-2} \\
=\left(n-a_{1}\right)^{a_{1}-1} \cdots\left(n-a_{k}\right)^{a_{k}-1} n^{k-2} .
\end{array}
$$

Conversely, for each sequence that meets the above conditions, find the smallest element not in the sequence, denoted as $b$, and assume $b \in A_{i}$.

If $\left|A_{i}\right|>1$, then establish a flight route between $b$ and the first element $c_{i, 1}$ of the $i$-th segment, and then delete $c_{i, 1}$ from the sequence;

If $\left|A_{i}\right|=1$, then establish a flight route between $b$ and the first element $c_{k+1,1}$ of the $(k+1)$-th segment, and then delete $c_{k+1,1}$ from the sequence.

Consider the remaining sequence as a sequence composed of elements from $S-\{b\}$, and repeat the above operation.
After $n-2$ operations, all elements in the sequence are deleted, and $S$ has two elements left. Establish a flight route between these two elements, thus establishing a total of $n-1$ flight routes. By induction, all cities can be connected by flight routes.

Since the above two operations are reversible, the number of flight route designs that meet the conditions is
$$
n^{k-2} \prod_{i=1}^{k}\left(n-a_{i}\right)^{a_{i}-1} \text {. }
$$
(Zhang Run, Xiao Jiagu, Chongqing Bashu High School, 400013)