The approach that you are referencing is meant to do the entire calculation in one gigantic "move", which I'm not a big fan of. It's simple enough to break the calculation into 3 "pieces" and then add together the results.
However, since you asked though, this calculation is based on two ideas:
1) You have to choose 3 people from a group of 10 (that's the first part of the calculation)
2) Since the group has to include at least 1 professor, you CAN'T have a group that is made up of 3 students.
Since there are 6 total students, there are 6!/3!3! different groups of 3 students that could be formed. We have to REMOVE those options from the prior part of the calculation. This gives us:
(All possible groups of 3) - (All groups of 3 that are JUST students) = (groups of 3 with at least 1 professor).
GMAT assassins aren't born, they're made,
EMPOWERgmat GMAT Club Page, Study Plans, & Discounts