Question

From a class of 12 boys and 10 girls, 10 students are to be chosen for a competition, including at least 4 boys and 4 girls. The 2 girls who won the prize last year should be included. In how many ways can the selection be made?

Answer 1

Two particular girls are always chosen.  So it remains that

From  12 boys          8 girls, ::    8 students to be chosen.
with  at least 4 boys and 2 girls.
total 4 boys, 6 girls, or 5 boys 5 girls or 6 boys and 4 girls... possible.

number of ways:  12C4 * 8C4 + 12C5 * 8C3 + 12C6  * 8C2 
    = 12*11*10*9*8*7*6*5/(24 *24) + 12*11*10*9*8*8*7*6/(120*6)
      + 12*11*10*9*8*7*8*7/(720*2)
   = 12*11*10*9*7 [5/12  + 8/15 + 14/45 ] = 462 * [75+96+56]
   = 462 * 227