Assign 4 brothers to room A in C(12,4)=12!/(4!8!) ways
Assign 6 brothers to room B in C(8,6)=8!/(6!2!) ways
Assign 2 brothers to room C in C(2,2)=2!(2!0!) ways
Total number of ways is the product
C(12,4)*C(8,6)*C(2,2)=12!8!2!/(4!8!6!2!2!0!)=12!/(4!6!2!)=13860 ways
Incidentally, this is the same answer as the number of permutations of arranging 12 objects in a line, with 6,4 and 2 objects being identical
P(4,6,2)=12!/(4!6!2!)=13860.
Think about how this relates to the problems of the brothers.
