I got 645 too..
the way I solved this problem was to take the combinations where there would be no professors in the group and subtract it from the total no. of combinations-
7C3=35-no prof at all
Good job. Always look for "complements" to save you lots of time. The GMAT does this a lot.
Former Senior Instructor, Manhattan GMAT and VeritasPrep
Vice President, Midtown NYC Investment Bank, Structured Finance IT
MFE, Haas School of Business, UC Berkeley, Class of 2005
MBA, Anderson School of Management, UCLA, Class of 1993