Can you please tell me why my method is wrong....
(10/28) * (8/27 * 7/26) * (10/25)
This answer assumes that you are selecting a Man, two Women and a Child in that order. i.e. MWWC. In other words, you are introducing a restriction that is not in the original problem.
Your answer does not account for additional selections which are possible arrangements of "MWWC" itself. Note that "MWWC" can be arranged in 4!/2! ways = 4*3 = 12 ways.
Note that the difference between the original answer and your answer is just this - 4*3 = 12.
Just by rearranging the terms, your answer can also be expressed as (10C1 * 8C2 * 10C1 / 28C4) * (4*3).
When the order is not important, you just select 1 man from 10, 2 women from 8 and 1 children from 10 and multiply.
Even though it really depends on the problem, a general rule of thumb, the approach you have taken is more appropriate for problems involving a deck of cards or problems involving picking blue/red/black marbles from a bag/box or problems involving picking a blue/red/black sox from a bag/box etc.