Example 5 Let $A=\{1,2,3, \cdots \cdots, n\},\left\{a_{1}, a_{2}, a_{3}\right\}\left(a_{1}<a_{2}<a_{3}\right)$ be a three-element subset of $A$, if it satisfies $a_{3} \geqslant a_{2}+3 \geqslant a_{1}+6$, then $\left\{a_{1}, a_{2}, a_{3}\right\}$ is called a “good subset” of $A$. Find the number of “good subsets” of $A$.
Solution: In the sequence $1,2,3, \cdots \cdots, n$, select three numbers. After selecting a number, insert a "board" right after this number. The three "boards" divide $1,2,3, \cdots \cdots, n$ into four parts, with the number of elements in each part from left to right being $x_{1}, x_{2}, x_{3}, x_{4}$, respectively. To make the selected three numbers $\left\{a_{1}, a_{2}, a_{3}\right\}$ form a "good subset", i.e., $a_{3} \geqslant a_{2}+3, a_{2} \geqslant a_{1}+3$, it is necessary that $x_{1} \geqslant 1, x_{2} \geqslant 3, x_{3} \geqslant 3 ; x_{4} \geqslant 0$. Therefore, the number of "good subsets" of $A$ is equal to the number of integer solutions to the equation $x_{1}+x_{2}+x_{3}+x_{4}=n$ that satisfy $x_{1} \geqslant 1, x_{2} \geqslant 3, x_{3} \geqslant 3, x_{4} \geqslant 0$. Hence, we can set $x^{\prime}{ }_{1}=x_{1} \geqslant 1, x^{\prime}{ }_{2}=x_{2}-2 \geqslant 1, x^{\prime}{ }_{3}=x_{3}-2 \geqslant 1, x^{\prime}{ }_{4}=x_{4}+1 \geqslant 1$. The number of integer solutions to the equation $x_{1}+x_{2}+x_{3}+x_{4}=n$ that satisfy $x_{1} \geqslant 1, x_{2} \geqslant 3, x_{3} \geqslant 3, x_{4} \geqslant 0$ is equal to the number of positive integer solutions to the equation $x^{\prime}{ }_{1}+x^{\prime}{ }_{2}+x^{\prime}{ }_{3}+x^{\prime}{ }_{4}=n+3$, and the number of positive integer solutions to this equation is $C_{n+2}^{3}$. Therefore, the number of "good subsets" of set $A$ is $C_{n+2}^{3}$.