2. If numbers $a_{1}, a_{2}, a_{3}$ are taken in increasing order from the set $1,2,3, \cdots, 14$, such that $a_{2}-a_{1} \geqslant 3, a_{3}-a_{2} \geqslant 3$. Find the number of all different ways to choose such numbers.
(Generally, we solve the following problem:
Find the number of different ways to choose $r$ numbers $a_{1}, a_{2}, \cdots, a_{r}$ from the set $\{1,2, \cdots, n\}$ that satisfy the following conditions:
(1) $1 \leqslant a_{1}<a_{2}<\cdots<a_{r} \leqslant n$;
(2) $a_{k+1}-a_{k} \geqslant m, k=1,2, \cdots, r-1$, where $m \in \mathbf{N}$.
Let the set of all $r$-tuples that satisfy (1) and (2) be denoted by $A$. From (2), we have $a_{k} \leqslant a_{k+1}-m(k=1,2, \cdots, r-1)$, i.e., $a_{k}<a_{k+1}-$ $(m-1)$. Therefore,
$$
\begin{array}{l}
a_{1}<a_{2}-(m-1)<a_{3}-2(m-1) \\
<a_{4}-3(m-1)<\cdots<a_{r}-(r-1)(m-1) . \\
\text { Let } b_{i}=a_{i}-(i-1)(m-1), i=1,2, \cdots, r, \text { then } \\
1 \leqslant b_{1}<b_{2}<\cdots<b_{r} \leqslant n-(r-1)(m-1) .
\end{array}
$$

Let the set of (unordered) $r$-tuples from the set $\{1,2, \cdots, n-(r-1)(m-1)\}$ be denoted by $B$, then there is a one-to-one correspondence between $A$ and $B$.
$$
\text { Hence }|A|=\mathrm{C}_{n-(r-1)(m-1) .}^{r} \text { ) }
$$