1. Problem a: Find the number of committees of 5 members from 8 women and 4 men with at most 2 men. - "At most 2 men" means 0, 1, or 2 men. - Total members = 5. Calculate for each case: - Case 0 men: Choose 0 men from 4 and 5 women from 8: $$\binom{4}{0} \times \binom{8}{5} = 1 \times 56 = 56$$ - Case 1 man: Choose 1 man from 4 and 4 women from 8: $$\binom{4}{1} \times \binom{8}{4} = 4 \times 70 = 280$$ - Case 2 men: Choose 2 men from 4 and 3 women from 8: $$\binom{4}{2} \times \binom{8}{3} = 6 \times 56 = 336$$ Total committees: $$56 + 280 + 336 = 672$$ 2. Problem b: Write the total number of permutations of a set of $n$ objects arranged in a circle. - The number of distinct circular permutations of $n$ objects is: $$ (n-1)! $$ 3. Problem c: Eight members sit at a round table. President and vice president must sit together. - Treat president and vice president as a single unit. - Then we have 7 units to arrange around the table. - Number of circular permutations of 7 units: $$ (7-1)! = 6! = 720 $$ - President and vice president can switch seats within their unit: $$ 2! = 2 $$ Total arrangements: $$ 720 \times 2 = 1440 $$ Final answers: - a) 672 committees - b) $ (n-1)! $ permutations - c) 1440 seating arrangements