Answer:
If order matters: 2,594,592,000 ways.
If order does not matter: 64,350 ways
Step-by-step explanation:
Assuming that the order matters when picking the non-pitching positions (since they are different positions), the number of possible different starting lineups is given by the permutation of picking 8 players out of 15, multiplied by 10 (pick one out of 10 pitchers):
Now if the order of the field players is not important, the number of possible starting lineups is given by the combination of picking 8 players out 15, multiplied by 10:
Therefore, the number of ways to pick the starting lineup is:
If order matters: 2,594,592,000 ways.
If order does not matter: 64,350 ways