Choosing two people from 2 boys and 2 girls 
If you have 2 boys and 2 girls, how many ways are there to choose two
  people?

One possible answers comes from saying that there are two possible genders for the first person and two possible genders for the second, giving $2 \cdot 2 = 4$ ways.
However, this can also be done as $\binom{4}{2}$ giving 6 ways.
Why don't these two answers match?
 A: 

If you have 2 boys and 2 girls, how many ways are there to choose two people?

One possible answers comes from saying that there are two possible genders for the first person and two possible genders for the second, giving $2\cdot 2=4$ ways.
However, this can also be done as ${4\choose 2}$ giving $6$ ways.
Why don't these two answers match?

The first way merely counts the ways to select the gender of the persons, not their identity.  It's also wrong, as there are only 3 ways to do so ${BB}, {BG}, {GG}$.  In addition to this you must still select the identity of each selected gender; you neglected this step.  There's one way to select two boys, four ways to select a boy and a girl, and one ways to select two girls; for a total of six ways.
The second method directly counts ways to select the identity of the persons, and correctly disregards the order of selection.
A: If we want to count the number of ways to have two persons in which $1$ must be a girl and $1$ must be a boy then there are: $2\cdot 2 = 4$ ways.
If we want to count the number of ways to have two persons then we have more possibilities since we could have two boys and two girls. So adding $2$ to the above answer we have: $4 + 2 = 6$ ways.
