21.3.13 $\star \star$ Before the World Cup, the coach of country $F$ prepared to observe the seven players $A_{1}, A_{2}, \cdots, A_{7}$ by having them play in three training matches (each 90 minutes). Assuming that at any moment during the matches, exactly one of these players is on the field, and that the total playing time (in minutes) of $A_{1}, A_{2}, A_{3}, A_{4}$ is divisible by 7, and the total playing time (in minutes) of $A_{5}, A_{6}, A_{7}$ is divisible by 13. If the number of substitutions per match is unlimited, how many different scenarios are there for the total playing time of each player?
Let the playing time of the $i$-th player be $x_{i}$ minutes $(i=1,2, \cdots, 7)$. The problem is transformed into finding the positive integer solutions of the indeterminate equation
$$
x_{1}+x_{2}+\cdots+x_{7}=270
$$

satisfying the conditions $7 \mid x_{i}(i=1,2,3,4)$, and $13 \mid x_{j}(j=5,6,7)$. Let $x_{1}+x_{2}+x_{3}+x_{4}=7 m, x_{5}+x_{6}+x_{7}=13 n$, then $7 m+13 n=270$, and $m$, $n \in \mathbf{N}^{+}, m \geqslant 4, n \geqslant 3$.
It is easy to find that the positive integer solutions $(m, n)$ satisfying the above conditions are
$$
(m, n)=(33,3),(20,10),(7,17) .
$$

When $(m, n)=(33,3)$, $x_{5}=x_{6}=x_{7}=13$. Let $x_{i}=7 y_{i}(i=1,2,3,4)$, then $y_{1}+y_{2}+y_{3}+y_{4}=33$, there are $\mathrm{C}_{33-1}^{4-1}=\mathrm{C}_{32}^{3}=4960$ groups of positive integer solutions $\left(y_{1}, y_{2}, y_{3}, y_{4}\right)$, at this time (1) has 4960 groups of positive integer solutions satisfying the conditions.

When $(m, n)=(20,10)$, let $x_{i}=7 y_{i}(i=1,2,3,4), x_{j}=13 y_{j}(j=5,6,7)$, then
$$
y_{1}+y_{2}+y_{3}+y_{4}=20, y_{5}+y_{6}+y_{7}=10,
$$

at this time (1) has $\mathrm{C}_{19}^{3} \times \mathrm{C}_{9}^{2}=31884$ groups of positive integer solutions satisfying the conditions.
When $(m, n)=(7,17)$, let $x_{i}=7 y_{i}(i=1,2,3,4), x_{j}=13 y_{j}(j=5,6,7)$, then
$$
y_{1}+y_{2}+y_{3}+y_{4}=7, y_{5}+y_{6}+y_{7}=17,
$$

at this time (1) has $\mathrm{C}_{6}^{3} \times \mathrm{C}_{16}^{2}=2400$ groups of positive integer solutions satisfying the conditions.
In summary, the number of groups of positive integer solutions of (1) satisfying the conditions is
$$
4960+34884+2400=42244 .
$$