Example 7 (1989 National High School League Question) If numbers are taken from $1,2, \cdots, 14$ in ascending order as $a_{1}, a_{2}, a_{3}$, such that $a_{2}-a_{1} \geqslant 3, a_{3}-a_{2} \geqslant 3$, then the number of different ways to satisfy the above conditions is $\qquad$ kinds.
Solve: Fill in $C_{10}^{3}$. Reason: Establish the correspondence $\left(a_{1}, a_{2}, a_{3}\right) \rightarrow\left(a_{1}, a_{2}-2, a_{3}-4\right)$, then $a_{1}<a_{2}-2<a_{3}-4$ is selecting 3 numbers from $\{1,2, \cdots, 10\}$, which has $C_{10}^{3}$ ways. Since the above correspondence is clearly a one-to-one correspondence, therefore, the total number of different selections is $C_{10}^{3}$.