How many distinct permutations can be formed using the letters of the word "BASEBALL"?
There are $\square$ distinct permutations.
(Type a whole number.)

Step 1 ：The word 'BASEBALL' has 8 letters, with the letter 'B' appearing twice, the letter 'A' appearing twice, and the letter 'L' appearing twice.

Step 2 ：The number of distinct permutations of a multiset is given by the formula: \[\frac{n!}{n_1!n_2!...n_k!}\] where n is the total number of items, and each n_i is the number of times a particular item appears.

Step 3 ：In this case, n = 8, n_1 = n_2 = n_3 = 2 (for the letters 'B', 'A', and 'L'), and all other n_i = 1 (for the letters 'S' and 'E').

Step 4 ：Substituting these values into the formula, we get the number of distinct permutations as 5040.

Step 5 ：Final Answer: There are \(\boxed{5040}\) distinct permutations.