Selecting a Team with Captain Constraints from Debate Club

To determine the total number of ways to form the team with the given constraints, we break down the problem based on the condition that the team must contain at most one boy.

The debate club consists of:

• Total number of girls = $6$
• Total number of boys = $4$
• Team size = $4$ members (out of which $1$ member is designated as the captain)

Since the team can include at most one boy, we have two mutually exclusive cases:

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Case 1: The team contains $0$ boys and $4$ girls

1. Selecting the members: Choose $4$ girls out of $6$: $\binom{6}{4} = \frac{6 \times 5}{2 \times 1} = 15 \text{ ways}$

2. Selecting the captain: Choose $1$ captain from the $4$ selected team members: $\binom{4}{1} = 4 \text{ ways}$

Total ways for Case 1: $N_1 = 15 \times 4 = 60$

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Case 2: The team contains $1$ boy and $3$ girls

1. Selecting the members:

   • Choose $1$ boy out of $4$: $\binom{4}{1} = 4$ ways
   • Choose $3$ girls out of $6$: $\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$ ways

   Ways to select the members = $4 \times 20 = 80 \text{ ways}$

2. Selecting the captain: Choose $1$ captain from the $4$ selected team members: $\binom{4}{1} = 4 \text{ ways}$

Total ways for Case 2: $N_2 = 80 \times 4 = 320$

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Total Number of Ways

Adding the total ways from both cases gives: $\text{Total Ways} = N_1 + N_2 = 60 + 320 = 380$

Thus, the correct option is (A).