Permutation and Combination /
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Q69
2015,
Permutation and Combination
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?
1. Correct Answer: Option C: 24 2. Explanation: To solve this problem, we need to assign 5 tasks to 5 persons with certain constraints. Let's break it down step by step: ● Task-1 cannot be assigned to either person-1 or person-2. This means Task-1 can only be assigned to person-3, person-4, or person-5. So, there are 3 possible choices for Task-1.
● Task-2 must be assigned to either person-3 or person-4. This gives us 2 possible choices for Task-2.
○ After assigning Task-1 and Task-2, we will have 3 tasks left and 3 persons left. These remaining tasks can be assigned to the remaining persons in 3! (3 factorial) ways, which is 6 ways. Now, let's calculate the total number of ways to assign the tasks: ○ For Task-1, we have 3 choices. ○ For Task-2, we have 2 choices. ○ For the remaining tasks, we have 6 ways to assign them. Therefore, the total number of ways to assign the tasks is: \[ 3 \times 2 \times 6 = 36 \] However, we must consider that Task-2's assignment to person-3 or person-4 is independent of Task-1's assignment. Thus, we need to adjust for overcounting: ○ If Task-1 is assigned to person-3, Task-2 can only be assigned to person-4. ○ If Task-1 is assigned to person-4, Task-2 can only be assigned to person-3. ○ If Task-1 is assigned to person-5, Task-2 can be assigned to either person-3 or person-4. This gives us: ○ 1 way for Task-1 to person-3 and Task-2 to person-4. ○ 1 way for Task-1 to person-4 and Task-2 to person-3. ○ 2 ways for Task-1 to person-5 and Task-2 to either person-3 or person-4. Therefore, the correct calculation is: \[ (1 + 1 + 2) \times 6 = 4 \times 6 = 24 \] Thus, the correct answer is Option C: 24.