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 ?
- (a) 6
- (b) 12
- (c) 24 ✓ UPSC's answer
- (d) 144
Why the answer is (c)
• Task-2 must be assigned to either Person-3 or Person-4, giving 2 possible choices for this specific task.
• Once Task-2 is assigned, 4 persons remain for the remaining 4 tasks (Task-1, Task-3, Task-4, Task-5).
• Task-1 cannot be assigned to Person-1 or Person-2. We must check if Person-1 or Person-2 are still available.
• Case 1: Task-2 is assigned to Person-3. Remaining persons: 1, 2, 4, 5. Task-1 cannot go to 1 or 2, so it must go to 4 or 5 (2 choices). The remaining 3 persons can be assigned to the last 3 tasks in 3! = 6 ways. Total for Case 1: 2 * 6 = 12 ways.
• Case 2: Task-2 is assigned to Person-4. Remaining persons: 1, 2, 3, 5. Task-1 cannot go to 1 or 2, so it must go to 3 or 5 (2 choices). The remaining 3 persons can be assigned to the last 3 tasks in 3! = 6 ways. Total for Case 2: 2 * 6 = 12 ways.
• Total ways = 12 + 12 = 24. Thus, option (c) is correct.
Why the other options are wrong
- (a) 6
- 6 is incorrect because it fails to account for the permutations of the remaining three tasks after assigning Task-2 and Task-1.
- (b) 12
- 12 is incorrect because it only accounts for one of the two possible assignments for Task-2 (either Person-3 or Person-4).
- (d) 144
- 144 is incorrect because it ignores the constraints on Task-1 and Task-2, effectively calculating 4! * 3! or similar unrestricted permutations.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2015, held on 23 August 2015. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.