Some SAT questions, like this one, require us to count possibilities. In this case, we're asked to count "how many numbers can be formed" based on the rules described. We'll want avoid attempting to list all possible arrangements of the digits 1, 2, 3, 4 and 5, so we can look for a pattern.
One way to analyze the pattern here is to consider the number of ways of filling each position:
( 5 ) ( 4 ) ( 3 ) ( 2 ) ( 1 )
Shown above are the number of possible entries in each position. For the first position, we have 5 possible entries: the numbers 1, 2, 3, 4 and 5. Say we chose any one of those, then our possible entries for the second position will just be four numbers, since we cannot repeat any numbers. Since we will have to choose one again for the third position, then our possible entries would be down to three numbers, for the fourth position we'll only have two number options, and for the fifth and final position, there would only be one number left as an entry. Now by multiplying all of those, we get 120.
The correct answer is (E).