1. **State the problem:** We have 12 people and want to form a committee of 5. From this committee, we select a President and a Secretary (distinct roles). Two people, X and Y, refuse to be on the committee together. We need to find the number of possible ordered role-assignments (committee + president + secretary) satisfying this restriction. 2. **Total ways without restriction:** - Number of ways to choose any 5-person committee from 12: $$\binom{12}{5}$$ - For each committee, number of ways to assign President and Secretary from the 5 members: $$5 \times 4 = 20$$ - Total without restriction: $$\binom{12}{5} \times 20$$ 3. **Count committees where X and Y are both included (not allowed):** - Fix X and Y in the committee, choose remaining 3 from the other 10: $$\binom{10}{3}$$ - For each such committee, assign President and Secretary: $$5 \times 4 = 20$$ - Total disallowed: $$\binom{10}{3} \times 20$$ 4. **Calculate values:** - $$\binom{12}{5} = \frac{12!}{5!7!} = 792$$ - $$\binom{10}{3} = \frac{10!}{3!7!} = 120$$ 5. **Calculate allowed assignments:** $$\text{Allowed} = 792 \times 20 - 120 \times 20 = (792 - 120) \times 20 = 672 \times 20 = 13440$$ **Final answer:** $$\boxed{13440}$$ ordered role-assignments satisfy the restriction.