Question 1 Given positive integers $m, n$ satisfying $m \geqslant n \geqslant 2$. Find the number of injections $f:\{1,2, \cdots, n\} \rightarrow\{1,2, \cdots, m\}$, such that there is a unique $i \in\{1,2, \cdots, n-1\}$, satisfying $f(i)>f(i+1)$. ${ }^{[1]}$
(2012, Romanian Mathematical Olympiad)
The original solution is rather complicated, this article provides a simpler solution.
Let the image set of the mapping $f$ be
$$
A=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\} .
$$

Then there are $\mathrm{C}_{m}^{n}$ such image sets.
Without loss of generality, let $A=\{1,2, \cdots, n\}$.
Next, we calculate the number of one-to-one mappings $f$:
$$
\{1,2, \cdots, n\} \rightarrow\{1,2, \cdots, n\}
$$

such that there is a unique $i \in\{1,2, \cdots, n-1\}$ satisfying $f(i)>f(i+1)$.
Then $f(1)f(i+1)$, so we need to exclude the case
$$
f(1)=1, f(2)=2, \cdots, f(i)=i
$$

Therefore, the number of one-to-one mappings $f$:
$$
\begin{array}{r}
\{1,2, \cdots, n\} \rightarrow\{1,2, \cdots, n\} \\
\text { is } \sum_{i=1}^{n-1}\left(\mathrm{C}_{n}^{i}-1\right)=2^{n}-n-1 \text { . }
\end{array}
$$

Thus, the number of required injections is $\mathrm{C}_{m}^{n}\left(2^{n}-n-1\right)$.
Make changes to question 1.