Since choosing Tom, Dick, and Harry is the same as choosing Dick, Harry, and
Tom, order doesn't matter, so we use combinations, not permutations.

Choose the workers for job site A as any 3 of the 12 in C(12,3)=220 ways.
Choose the workers for job site B as any 4 of the remaining 9 in C(9,4)=126 ways.
Choose the workers for job site C as the remaining 5 in C(5,5)=1 way. 

Answer: 220×126×1 = 27720 ways

Notice that it does not matter in which order we choose the workers for the
three sites. For instance suppose we chose workers for C first, then A, then B:

Choose the workers for job site C as any 5 of the 12 in C(12,5)=792 ways.
Choose the workers for job site A as any 3 of the remaining 7 in C(7,3)=35 ways.
Choose the workers for job site B as the remaining 4 in C(4,4)=1 way. 

Answer: 792×35×1 = 27720 ways

Although we multiplied different numbers together, we get the same
answer 27720 both times.  You can try other picking orders and you'll 
see you will always get 27720 in the end.

Edwin