Example 1-9 A string of length 5 is formed by 26 English letters, requiring $(1) 6$ vowels $a, e, i, o, u, y$ not to be adjacent (2) the remaining 20 consonants do not have 3 adjacent; (3) adjacent consonants are not the same. Try to find how many such strings there are.
Let $\alpha$ represent the set of $a, e, i, o, u, y$, and $\beta$ represent the set of the remaining 20 consonants. According to the requirements, the structure of such strings can be expressed as:
According to the multiplication rule:
(1) $\alpha \beta \beta \alpha \beta$ type
$$
n_{1}=6 \times 20 \times 19 \times 6 \times 20=6^{2} \times 20^{2} \times 19 ;
$$
(2) $\alpha \beta \alpha \beta \alpha$ type
$$
n_{2}=6 \times 20 \times 6 \times 20 \times 6=6^{3} \times 20^{2} ;
$$
(3) $\alpha \beta \alpha \beta \beta$ type
$$
n_{3}=6 \times 20 \times 6 \times 20 \times 19=6^{2} \times 20^{2} \times 19 ;
$$
(4) $\beta \beta \alpha \beta \alpha$ type
$$
n_{4}=20 \times 19 \times 6 \times 20 \times 6=6^{2} \times 20^{2} \times 19 ;
$$
(5) $\beta \beta \alpha \beta \beta$ type
$$
n_{5}=20 \times 19 \times 6 \times 20 \times 19=6 \times(20 \times 19)^{2} ;
$$
(6) $\beta \alpha \beta \alpha \beta$ type
$$
n_{6}=20 \times 6 \times 20 \times 6 \times 20=6^{2} \times 20^{3} ;
$$
(7) $\beta \alpha \beta \beta \alpha$ type
$$
n_{7}=20 \times 6 \times 20 \times 19 \times 6=6^{2} \times 20^{2} \times 19 .
$$

According to the addition rule
$$
\begin{aligned}
n & =n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7} \\
& =6 \times 20^{2}\left[6 \times 19+6^{2}+6 \times 19+6 \times 19+19^{2}+6 \times 20+6 \times 19\right] \\
& =6 \times 20^{2}\left[24 \times 19+6^{2}+6 \times 20\right] \\
& =6 \times 20^{2} \times 973=2335200 .
\end{aligned}
$$