Answer
$2520$ batting orders.
Work Step by Step
Total: $3+4+1+1=9$ players.
It is a permutation of 3 outfielders, 4 infielders, 1 pitcher, and 1 catcher:
$\frac{9}{3!\times 4!\times 1!\times1!}=2520$
You can help us out by revising, improving and updating this answer.
Update this answerAfter you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.