1. **Problem statement:** A robotics team of 6 students is selected from 4 boys and 3 girls. Find: (a)(i) Number of different teams with no restrictions. (a)(ii) Number of different teams with at least 3 boys. (b) Number of ways to select and arrange all boys and 2 girls in a line so that all boys sit together. --- 2. **Step (a)(i): Number of ways with no restriction** Total students = 4 boys + 3 girls = 7 We want to choose 6 out of 7: $$\binom{7}{6} = 7$$ So, 7 different teams can be chosen. --- 3. **Step (a)(ii): Number of ways with at least 3 boys** We consider teams with exactly 3 boys and 3 girls, or 4 boys and 2 girls. - For 3 boys, choose $$\binom{4}{3} = 4$$ and for 3 girls choose $$\binom{3}{3} = 1$$ - For 4 boys, choose $$\binom{4}{4} = 1$$ and for 2 girls choose $$\binom{3}{2} = 3$$ Total teams: $$4 \times 1 + 1 \times 3 = 4 + 3 = 7$$ --- 4. **Step (b): Number of ways to select and arrange all boys and 2 girls in a line with boys together** - We must select all 4 boys and 2 girls from 3 girls: Number of ways to choose 2 girls: $$\binom{3}{2} = 3$$ - Treat the 4 boys as a single block to keep them together. Now, arrange this block plus 2 girls: Number of objects to arrange = 1 block + 2 girls = 3 Number of ways to arrange these 3 objects: $$3! = 6$$ - Inside the boys block, arrange 4 boys: $$4! = 24$$ - Total arrangements: $$3 \times 6 \times 24 = 432$$ --- **Final answers:** (a)(i) 7 teams (a)(ii) 7 teams (b) 432 ways