There are 5 tasks and 5 persons. Task-1 cannot be assigned to either person-1, or person-2. Task-2 must be assigned to either person-3 or person-4. Every person is to be assigned one task. In how many ways can the assignment be done?

The correct option is 24.

We are given 5 tasks (Task-1, Task-2, Task-3, Task-4, Task-5) and 5 persons (Person-1, Person-2, Person-3, Person-4, Person-5). Each person must be assigned exactly one task. We need to find the number of valid assignments under the following constraints:
1. Task-1 cannot be assigned to Person-1 or Person-2.
2. Task-2 must be assigned to either Person-3 or Person-4.

Let us analyze the assignments step-by-step. Since Task-2 has a strict constraint specifying it must go to either Person-3 or Person-4, we can split the problem into two mutually exclusive cases based on who receives Task-2.

Case 1: Task-2 is assigned to Person-3
If Task-2 is assigned to Person-3, then Person-3 is no longer available for any other task. Now we need to assign the remaining tasks to the remaining 4 persons (Person-1, Person-2, Person-4, Person-5).
Let's assign Task-1 next, since it has restrictions. Task-1 cannot be assigned to Person-1 or Person-2. Therefore, Task-1 must be assigned to either Person-4 or Person-5.
There are 2 choices for Task-1 (either Person-4 or Person-5).
Once Task-1 is assigned to one of these 2 people, we have 3 tasks left (Task-3, Task-4, Task-5) and 3 persons remaining. These 3 tasks can be assigned to the remaining 3 persons in any order.
The number of ways to assign the remaining 3 tasks is:
$3!=6$ ways.
Thus, the total number of ways for Case 1 is:
$2\times 6=12$ ways.

Case 2: Task-2 is assigned to Person-4
If Task-2 is assigned to Person-4, then Person-4 is no longer available. We need to assign the remaining tasks to the remaining 4 persons (Person-1, Person-2, Person-3, Person-5).
Again, Task-1 cannot be assigned to Person-1 or Person-2. Thus, Task-1 must be assigned to either Person-3 or Person-5.
There are 2 choices for Task-1 (either Person-3 or Person-5).
After assigning Task-1, we have 3 tasks left (Task-3, Task-4, Task-5) and 3 persons remaining. The remaining 3 tasks can be assigned to the 3 remaining persons in:
$3!=6$ ways.
Thus, the total number of ways for Case 2 is:
$2\times 6=12$ ways.

Total Number of Ways:
Since Case 1 and Case 2 are mutually exclusive and cover all possibilities for the assignment of Task-2, we add the number of ways from both cases:
Total ways = Ways from Case 1 + Ways from Case 2
$\text{Total ways}=12+12=24$

Therefore, the assignment can be done in 24 ways.