Question

There are 5 rotten apples in a crate of 25 apples. How many samples of 3 can
be drawn from the crate?

Answer 1

There are 25!/[3!*22!) = 25*24*23/6 = 23,000 combinations of ways that three apples can be drawn, regardless of how many are rotten.

Are you sure you are asking the right question? If you want the probability that one is rotten in a sample of three, that would be
(5/25)*(22/24)(21/23)*3 = 50.2%
The probability that none of the three drawn (at random) are rotten is 49.6%