A debate club consists of $6$ girls and $4$ boys. A team of $4$ members is to be selected from this club including the selection of a captain (from among these $4$ memers) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

$\text{Given: } 6 \text{ girls},\ 4 \text{ boys}$

$\text{Team: } 4 \text{ members} + 1 \text{ captain (from the } 4)$

$\text{Condition: at most one boy}$

$\text{To find: number of ways of selecting the team}$

$\text{Plan: count valid teams (0 boy / 1 boy), then the captain}$

$\text{Captain: } 1 \text{ from the team of } 4 \Rightarrow 4 \text{ ways per team}$

$\text{Total} = 4 \times (\#\text{valid teams})$

$\text{Case 0 boys: } \binom{6}{4} = 15$

$\text{Case 1 boy: } \binom{4}{1}\binom{6}{3} = 4 \times 20 = 80$

$\#\text{valid teams} = 15 + 80 = 95$

$\text{Each team} \to 4 \text{ captain choices}$

$\text{Total} = 95 \times 4 = 380$

$\text{Check (captain first): } 6\left[\binom{5}{3}+\binom{4}{1}\binom{5}{2}\right] = 6(10+40)=300$

$4\binom{6}{3}=4\times 20=80 \Rightarrow 300+80=380$

$\boxed{380}\quad\text{Option (1)}$