Example 4. If from the numbers $1,2, \cdots, 141$, we sequentially select $a_{1}, a_{2}, a_{2}$, such that $a_{2}-a_{1} \geqslant 3, a_{3}-a_{2} \geqslant 3$, then how many different ways of selection are there that meet the above requirements?
Let $S=\{1,2, \cdots, 14\}, S^{\prime}=\{1, 2, \cdots, 10\}$. Suppose $\left\{a_{1}, a_{2}, a_{3}\right\}$ is a subset of $S$ and satisfies $a_{1}^{\prime}=a_{1}, a_{2}^{\prime}=a_{2}-2, a_{3}^{\prime}=a_{3}-4$, i.e., the following mapping is established:
$$
\begin{array}{c}
\left(a_{1}, a_{2}, a_{3}\right) \rightarrow\left(a_{1}^{\prime}, a_{2}^{\prime}, a_{3}^{\prime}\right) \\
=\left(a_{1}, a_{2}-2, a_{3}-4\right) .
\end{array}
$$
The number of ways to choose such subsets is equal to the number of ways to choose any 3 different numbers from $S^{\prime}$, i.e., $C_{10}^{3}=120$.