Correct Option (3)
The problem involves assigning 5 tasks to 5 persons with specific constraints. The number of ways to perform this assignment can be determined by considering each constraint sequentially.
- Assignment for Task 2: Task 2 must be assigned to either Person 3 or Person 4.
- This provides 2 possible ways for Task 2.
- Assignment for Task 1: Task 1 cannot be assigned to Person 1 or Person 2. The potential assignees are Person 3, Person 4, or Person 5.
- Since one person (either Person 3 or Person 4) has already been assigned Task 2, there are 2 persons remaining from the set {Person 3, Person 4, Person 5} who can be assigned Task 1.
- This provides 2 possible ways for Task 1.
- Assignment for the Remaining Tasks (Tasks 3, 4, 5):
- Two persons have now been assigned Task 1 and Task 2.
- This leaves 3 remaining persons and 3 remaining tasks.
- These 3 tasks can be assigned to the 3 remaining persons in 3! (3 factorial) ways.
- 3! = 3 × 2 × 1 = 6 ways.
- Total Number of Ways: To find the total number of distinct assignment ways, multiply the possibilities from each step:
- Total ways = (Ways for Task 2) × (Ways for Task 1) × (Ways for Remaining Tasks)
- Total ways = 2 × 2 × 6 = 24.
Therefore, the assignment can be done in 24 ways.
Incorrect Options:
Options 1 (6), 2 (12), and 4 (144) are incorrect because they do not account for all the specified constraints and the systematic application of combinatorial principles required to determine the total number of unique task assignments. The correct calculation, considering all conditions, yields 24 ways.