Final answer:
To determine the number of 9-player batting orders for a baseball team with 18 players, we calculate the permutation of 18 items taken 9 at a time, resulting in 1,307,674,368,000 possible orders.
Step-by-step explanation:
The question asks how many 9-player batting orders are possible for a baseball team with 18 players. This is a problem of permutations, where order matters. To find this, we use the formula for permutations without repetition, which is n!/(n-r)!, where 'n' is the total number of items to choose from, and 'r' is the number of items being chosen.
In this case, we have n = 18 players to choose from and we want to create ordered line-ups of r = 9 players. Using the permutation formula:
P(18, 9) = 18! / (18-9)! = 18! / 9! = 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 = 1,307,674,368,000 possible 9-player batting orders.