The problem asks for the number of ways to form a 4-member team from a club of 6 girls and 4 boys, and then select a captain from the team, with the condition that the team has at most one boy.

The condition "at most one boy" means the team can have either zero boys or one boy. We can solve this by considering these two cases separately and then adding the number of ways for each case.

Case 1: The team has 0 boys and 4 girls.

1. Select the team members: We need to select 4 girls from the available 6 girls. The number of ways to do this is given by the combination formula $C(n, k) = \frac{n!}{k!(n-k)!}$. Number of ways to select 4 girls from 6 is: $C(6, 4) = C(6, 2) = \frac{6 \times 5}{2 \times 1} = 15 \text{ ways}$ We also need to select 0 boys from 4, which is $C(4, 0) = 1$ way. So, the total number of ways to select the group is $15 \times 1 = 15$.

2. Select the captain: From the selected team of 4 members (all girls), one member must be chosen as the captain. Number of ways to select 1 captain from 4 members is: $C(4, 1) = 4 \text{ ways}$

3. Total ways for Case 1: The total number of ways for this case is the product of the number of ways to select the team and the number of ways to select the captain. $\text{Total ways for Case 1} = 15 \times 4 = 60$

Case 2: The team has 1 boy and 3 girls.

1. Select the team members: We need to select 1 boy from the 4 boys and 3 girls from the 6 girls. Number of ways to select 1 boy from 4 is: $C(4, 1) = 4 \text{ ways}$ Number of ways to select 3 girls from 6 is: $C(6, 3) = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \text{ ways}$ The total number of ways to select the team members for this case is the product of these two values: $\text{Ways to select the team} = C(4, 1) \times C(6, 3) = 4 \times 20 = 80 \text{ ways}$

2. Select the captain: From the selected team of 4 members (1 boy and 3 girls), one member must be chosen as the captain. Number of ways to select 1 captain from 4 members is: $C(4, 1) = 4 \text{ ways}$

3. Total ways for Case 2: The total number of ways for this case is the product of the number of ways to select the team and the number of ways to select the captain. $\text{Total ways for Case 2} = 80 \times 4 = 320$

Total Number of Ways

The total number of ways of selecting the team is the sum of the ways from Case 1 and Case 2. $\text{Total ways} = (\text{Ways for Case 1}) + (\text{Ways for Case 2})$ $\text{Total ways} = 60 + 320 = 380$

Comparing this result with the given options: A: 380 B: 320 C: 260 D: 95

The calculated value 380 matches option A.