Counting property: We can arrange n unique items in n! ways
There are 5 unique letters in STUDY, so we can arrange those letters in 5! ways

5! = (5)(4)(3)(2)(1) = 120

Answer: 120

Cheers,
Brent
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Alternate approach:

Take the task of seating the 6 students and break it into stages.

Stage 1: Select a letter to be the FIRST letter
There are 5 letters to choose from, so we can complete stage 1 in 5 ways

Stage 2: Select a letter to be the SECOND letter
There are 4 letters remaining to choose from, so we can complete this stage in 4 ways

Stage 3: Select a letter to be the THIRD letter
There are 3 letters remaining to choose from, so we can complete this stage in 3 ways

Stage 4: Select a letter to be the FOURTH letter
There are 2 letters remaining to choose from, so we can complete this stage in 2 ways

Stage 5: Select a letter to be the LAST letter
There is 1 letter remaining to choose from, so we can complete this stage in 1 way

By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus seat all 5 letters) in (5)(4)(3)(2)(1) ways (= 120 ways)

Answer: 120

Note: the FCP can be used to solve the MAJORITY of counting questions on the GRE. So, be sure to learn

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Regards