MathDB

Before the FIFA world cup, the football coach of F country want to test seven players

A_1, A_2, \cdots, A_7

. He asks them to join in three training matches (90 minutes each), and everyone must appear in each match at least once. Suppose that at any moment during a match, one and only one of them enters the field, and the total time (measured in minutes) on the field for

A_1, A_2, A_3, A_4

are multiples of

7

and the total time for

A_5, A_6, A_7

are multiples of

13

. If the number of substitutions of players during each match is not limited, find the number of different cases. Note: If and only if the total time of a certian player is different, then the case is considered different.

Problem Statement