Question

There are 6 tasks and 6 persons. Task 1 cannot be assigned either to 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. 72 ways
• B. 192 ways
• C. 360 ways
• D. 144 ways
• E. 256 ways

Hint:

Use case analysis when constraints interact with each other.

Answer 1

The Correct Option is D

Step 1: Breaking Down the Problem:
We have 6 tasks (T1 to T6) and 6 persons (P1 to P6).
Condition 1: T1 cannot be assigned to P1 or P2. So T1 has 4 possible persons (P3, P4, P5, P6).
Condition 2: T2 must be assigned to either P3 or P4.

Step 2: Case Analysis:
Case 1: T1 is assigned to P3 or P4 (2 options).
Then T2 can be assigned to the other of P3/P4 (1 option remaining for T2 from P3, P4).
Remaining 4 tasks can be assigned to remaining 4 persons in \(4! = 24\) ways.
So Case 1 total = \(2 \times 1 \times 24 = 48\) ways.
Case 2: T1 is assigned to P5 or P6 (2 options).
Then T2 can be assigned to P3 or P4 (2 options).
Remaining 4 tasks can be assigned to remaining 4 persons in \(4! = 24\) ways.
So Case 2 total = \(2 \times 2 \times 24 = 96\) ways.

Step 3: Total Ways:
Total = \(48 + 96 = 144\) ways.