**Introduction:**

In this problem, we have three groups of children with different compositions of boys and girls. We need to calculate the probability of selecting one girl and two boys, where one child is selected at random from each group.

**Group 1: 3 girls and 1 boy**

Let's denote the children in this group as G1, G2, G3 (girls) and B1 (boy).

The probability of selecting one girl and two boys from this group can be calculated as follows:

- The probability of selecting a girl from this group is 3/4 (since there are 3 girls and 4 total children).
- The probability of selecting a boy from this group is 1/4 (since there is 1 boy and 4 total children).
- Since we need to select one girl and two boys, we can choose the girl in 3 ways and the boys in 2 ways (assuming the order of selection matters).
- Therefore, the total probability of selecting one girl and two boys from this group is (3/4) * (1/4) * 3 * 2 = 9/32.

**Group 2: 2 girls and 2 boys**

Let's denote the children in this group as G1, G2 (girls) and B1, B2 (boys).

The probability of selecting one girl and two boys from this group can be calculated as follows:

- The probability of selecting a girl from this group is 2/4 (since there are 2 girls and 4 total children).
- The probability of selecting a boy from this group is 2/4 (since there are 2 boys and 4 total children).
- Since we need to select one girl and two boys, we can choose the girl in 2 ways and the boys in 2 ways.
- Therefore, the total probability of selecting one girl and two boys from this group is (2/4) * (2/4) * 2 * 2 = 8/16.

**Group 3: 1 girl and 3 boys**

Let's denote the children in this group as G1 (girl) and B1, B2, B3 (boys).

The probability of selecting one girl and two boys from this group can be calculated as follows:

- The probability of selecting a girl from this group is 1/4 (since there is 1 girl and 4 total children).
- The probability of selecting a boy from this group is 3/4 (since there are 3 boys and 4 total children).
- Since we need to select one girl and two boys, we can choose the girl in 1 way and the boys in 3 ways.
- Therefore, the total probability of selecting one girl and two boys from this group is (1/4) * (3/4) * 1 * 3 = 3/16.

**Overall Probability:**

To find the overall probability, we need to consider the probabilities from all three groups. Since the selection from each group is independent, we can multiply the probabilities together.

The overall probability of selecting one girl and two boys is given by:

(Probability from Group 1) * (Probability from Group 2) * (Probability from Group 3) = (9/32) * (8/16) * (3/16) = 27/512.

Therefore