1.56 A sequence of length 15 composed of the letters $A, B$, satisfies the following conditions:

For any two consecutive letters, $A A$ should appear 5 times, $A B, B A, B B$ should each appear 3 times. How many such sequences are there?

For example, in $A A B B A A A A B A A B B B B$, $A A$ appears 5 times, $A B$ appears 3 times, $B A$ appears 2 times, and $B B$ appears 4 times, which does not meet the above conditions.
[Solution] First consider the sequences starting with $A$.
Since $A B, B A$ and $B B$ appear exactly 3 times, the structure of such sequences is
$$
\cdots A_{1} B_{1} \cdots A_{2} B_{2} \cdots A_{3} B_{3} \cdots A_{4} \text {. }
$$

Exactly 3 $B$s are inserted after $B_{i}(i=1,2,3)$. Let $x_{i}$ be the number of $B$s inserted after $B_{i}$, then
$$
x_{1}+x_{2}+x_{3}=3 \text {. }
$$

The number of non-negative integer solutions to this equation is $C_{3+2}^{2}=10$.
5 $A$s are inserted before $A_{i}(i=1,2,3,4)$. Let $y_{i}$ be the number of $A$s inserted before $A_{i}$, then
$$
y_{1}+y_{2}+y_{3}+y_{4}=5
$$

The number of non-negative integer solutions to this equation is $C_{8}^{3}=56$.
Therefore, the total number of such sequences is
$$
56 \times 10=560 \text { . }
$$