Example $7 m(m \geqslant 3)$ people $A_{1}, A_{2}, \cdots, A_{m}$ pass the ball to each other. Starting with $A_{1}$, this is called the 1st pass. After $n(n \geqslant 2)$ passes, the ball returns to $A_{1}$. Try to find the number of all different ways of passing the ball $N$.
Solution: Let the number of all possible ways for the ball to return to $A_{1}$ after $n$ passes be $c_{n}$. It is easy to see that
$$
c_{2}=m-1 \text {. }
$$

Since each player cannot pass the ball to themselves, they can only pass it to any one of the other $m-1$ players, so there are $m-1$ different directions for each pass.

According to the multiplication principle, after $n-1$ passes, there are $(m-1)^{n-1}$ different ways of passing, among which $c_{n-1}$ ways result in the ball returning to $A_{1}$, and the number of ways it ends up with other players is
$$
x_{n-1}=(m-1)^{n-1}-c_{n-1} \text { (ways). }
$$

Since the next pass cannot be to $A_{1}$ themselves, but any other player can pass to $A_{1}$, the number of ways to pass back to $A_{1}$ after $n$ passes is $x_{n-1} \leqslant c_{n}$; conversely, for each way the ball returns to $A_{1}$ after $n$ passes, removing the $n$-th pass, it must result in the ball being with someone other than $A_{1}$ after $n-1$ passes, so
$$
\begin{array}{l}
x_{n-1} \geqslant c_{n} . \\
\text { Thus, } c_{n}=x_{n-1} . \text { Therefore, } \\
c_{n}=(m-1)^{n-1}-c_{n-1} .
\end{array}
$$

From equations (4) and (5), we can solve it similarly to the coloring problem, obtaining
$$
c_{n}=\frac{1}{m}\left[(m-1)^{n}+(m-1)(-1)^{n}\right] \text {. }
$$

It can be seen that the recurrence relation of $b_{n}$ and $c_{n}$ is similar, and the expressions of $b_{n}$ and $c_{n}$ are also similar. The correspondence between them is:
(1) $n$ passes correspond to an $n$-sided polygon;
(2) $m$ players correspond to $m$ colors;
(3) the ball not being passed to oneself corresponds to adjacent points being colored differently;
(4) the ball being passed to only one person each time corresponds to each point being colored with only one color;
(5) in the coloring problem, there is no specified point $A_{1}$ that must be colored a certain color, which is equivalent to the ball not being specified to start from $A_{1}$ in the passing problem. Therefore, in the passing problem, if the ball can start from any player, the first pass is not 1 way but $m$ ways, and the recurrence relations and general formulas of the two problems are completely consistent.