Task assignment problem

Case 1: Task 2 is assigned to Person 3 or Person 4 (2 options). Suppose Task 2 is assigned to Person 3. Task 1 can be assigned to Person 4, 5, or 6 (3 options). Subcase 1a: Task 1 is assigned to Person 4. Remaining 4 tasks to 4 persons in $4! = 24$ ways. Total $= 2 \times 1 \times 1 \times 24 = 48$. Subcase 1b: Task 1 is assigned to Person 5 or 6 (2 options). Remaining 4 tasks in $4! = 24$ ways. Total $= 2 \times 2 \times 24 = 96$.

Total arrangements $= 48 + 96 = 144$.