$\qquad$ * Let $m, n \in \mathbf{N}_{+}, m \leqslant n$. Sets $A=\left\{a_{1}, \cdots, a_{m}\right\}, B=\left\{b_{1}, \cdots, b_{n}\right\}$.
(1) Find the number of all mappings from $A$ to $B$;
(2) Find the number of all injections from $A$ to $B$;
(3) Does there exist a surjection from $A$ to $B$?
(1) Since each element in $A$ has $n$ elements in $B$ that can serve as its image, the number of mappings from $A$ to $B$ is $n^{m}$.
(2) Determine the images of elements $a_{1}, \cdots, a_{m}$ in $A$ sequentially, with the number of methods being $n, n-1, \cdots, n-(m-$
1). Therefore, the number of injections from $A$ to $B$ is $n(n-1) \cdots(n-m+1)\left(=\mathrm{A}_{n}^{m}\right)$.
(3) When $n=m$, there exist surjective mappings from $A$ to $B$, and the number of surjective mappings is $n$!. When $n>m$, there are no surjective mappings from $A$ to $B$.