Assigning tasks to 5 persons with constraints

Handle the most heavily restricted task parameters first, branching using Task-2 options:

Now evaluate Task-1 choices. Neither P nor Q can take it, and R is already occupied with Task-2. The only remaining eligible persons for Task-1 are S or T. This yields exactly 2 choices. Once Task-1 and Task-2 are filled, 3 persons remain for the 3 unassigned tasks. They can be arranged in 3! = 6 ways. Total for Case 1 = 1 × 2 × 6 = 12 ways.

Now evaluate Task-1 choices. Neither P nor Q can take it, and S is occupied. The only available candidates left for Task-1 are R or T. This yields exactly 2 choices. The remaining 3 people fill the 3 remaining tasks in 3! = 6 ways. Total for Case 2 = 1 × 2 × 6 = 12 ways.

Correction Check on common intersections: Wait! Is there an overlap if T takes Task-1 in both cases? No, because the assignments for Task-2 are completely distinct (R vs S). Thus, adding the independent scenarios gives: 12 + 12 = 24 total ways.

Let's re-verify the step: if we choose to pick the people instead of task tracking, Task-2 has 2 options (R or S). Let's say it's R. S is free. Task-1 cannot be P or Q, so it must be S or T (2 options). Let's say it's S. Remaining people are P, Q, T for 3 tasks \rightarrow 3! = 6 ways. Total = 2 × 2 × 6 = 24 ways.

Answer: (d).