(1) exactly 2 teachers

   Choose the 2 teachers in C(6,2) ways.  That's 15 ways.
   Choose the 4 students in C(4,4) ways.  That's 1 way.

   Answer: 15*1 = 15

(2) at least 2 students.

  First find the number of committees consisting of any 6 of the 10 people
 
  Choose any 6 from the 10 people in C(10,6) ways.  That's 210 ways.   
      
  Then subtract the following from that number

       a. The number of committees with 1 student and 5 teachers

          Choose 1 student in C(4,1) ways.  That's 4 ways
          Choose 5 teachers in C(6,5) ways.  That's 6 ways

          That's 4*6 = 24 ways  

       b. The committee with 0 students and 6 teachers.

          Choose all 6 teachers.  That's C(6,6) or 1 way

Answer: 210 - 24 - 1 = 185 ways.

Edwin