Alex and 4 friends are playing baseball on the same team. Each time they play a game, they switch the batting order to a sequence that has not yet been used. What is the maximum number of games they can play before they have to repeat a batting order they have already used?

120.

Step-by-step explanation:

Since Alex and 4 friends are playing baseball on the same team, and each time they play a game, they switch the batting order to a sequence that has not yet been used, to determine what is the maximum number of games they can play before they have to repeat a batting order they have already used, the following calculation must be performed:

1 x 2 x 3 x 4 x 5 = X

2 x 3 x 4 x 5 = X

6 x 4 x 5 = X

24 x 5 = X

120 = X

Thus, the maximum number of games they can play before they have to repeat a batting order they have already used is 120.