There are 6 possible combinations of choosing 2 boys and 2 girls:
0 = girl
1 = boy

0011
0101
0110
1001
1010
1100

Each of these has a probability of 1/10
worked out by:
(3 x 3 x 2 x 2)/(6 x 5 x 4 x 3) = 1/10

6 x 1/10 = 3/5

A brief explanation of why each probability is 1/10:

Look at combination BBGG
p1 = p(1st choice is boy) = 3/6
p2 = p(2nd choice is boy) = 2/5
p3 = p(3rd choice is girl) = 3/4
p4 = p(4th choice is girl) = 2/3

p1 AND p2 AND p3 AND p4 = 3/6 * 2/5 * 3/4 * 2/3 = (3 * 2 * 3 * 2)/(6 * 5 * 4 * 3)

Note that for all combinations listed previously, the numerator will be the same (just multiplied in a different order). Also, the denominator will be the same.

The result is 36/360 = 1/10

Then 6 combinations * 1/10 = 3/5

There are 6 possible combinations of choosing 2 boys and 2 girls:
0 = girl
1 = boy

0011
0101
0110
1001
1010
1100

Each of these has a probability of 1/10
worked out by:
(3 x 3 x 2 x 2)/(6 x 5 x 4 x 3) = 1/10

6 x 1/10 = 3/5

There are 6 possible combinations of choosing 2 boys and 2 girls:
0 = girl
1 = boy

0011
0101
0110
1001
1010
1100

Each of these has a probability of 1/10
worked out by:
(3 x 3 x 2 x 2)/(6 x 5 x 4 x 3) = 1/10

6 x 1/10 = 3/5