Question: There are 6 tasks and 6 persons. Task 1 cannot be assigned either to person 1 or to 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?
- 144
- 180
- 192
- 360
- 716
Correct Answer: (A)
Solution and Explanation:
Approach Solution 1:
The problem statement informs that:
Given:
- There are 6 tasks T1, T2, T3, T4, T5, T6
- There are 6 people P1, P2, P3, P4, P5, P6.
- Task 1 cannot be assigned either to person 1 or to person 2
- Task 2 must be assigned to either person 3 or person 4.
- Every person is to be assigned one task.
Find out:
- In how many ways the task can be assigned?
Let’s consider that Task 2 is assigned to person 3, as stated in the problem statement.
If T2 is assigned to P3, there are 5 remaining tasks T1, T3, T4, T5, T6 and 5 remaining people P1, P2, P4, P5, P6.
T1 can't be assigned to P1 or P2, so T1 could be assigned to P4, P5, and P6.
T3 could be assigned to each of the 4 remaining people.
T4 could be assigned to each of the 3 remaining people.
T5 could be assigned to each of the 2 remaining people.
T6 could be assigned to the only remaining people.
Hence, the total combination, in this case, is 3*4*3*2*1.
If T2 is assigned to P4. The result is the same as the first case above. The total combination is 3*4*3*2*1
Hence, the number of ways the task can be assigned is 2*3*4*3*2*1=144
Approach Solution 2:
The problem statement informs that:
Given:
- There are 6 tasks T1, T2, T3, T4, T5, T6
- There are 6 people P1, P2, P3, P4, P5, P6.
- Task 1 cannot be assigned either to person 1 or to person 2
- Task 2 must be assigned to either person 3 or person 4.
- Every person is to be assigned one task.
Find out:
- the number of ways the task can be assigned.
Let’s take the task of assigning the tasks and break it into stages.
Let’s begin the sum with the most restrictive stage(s)
Stage 1: Select a person to complete task 2
Since task 2 must be allocated to either person 3 or person 4, then we can complete stage 1 in 2 ways
Stage 2: Select a person to complete task 1
There are 5 people remaining who have not been allocated a task
However, task 1 cannot be allocated to person 1 or to person 2
So, there are only 3 people to select from.
Therefore, we can complete stage 2 in 3 ways
Stage 3: Select a person to complete task 3
There are 4 people remaining, therefore, we can complete stage 3 in 4 ways
Stage 4: Select a person to complete task 4
There are 3 people remaining, therefore, we can complete stage 4 in 3 ways
Stage 5: Select a person to complete task 5
There are 2 people remaining, therefore, we can complete stage 5 in 2 ways
Stage 6: Select a person to complete task 6
There is 1 person remaining, therefore, we can complete stage 6 in 1 way
As per the Fundamental Counting Principle (FCP), we can meet all 6 stages (and thus allocate all 6 tasks) in (2)(3)(4)(3)(2)(1) ways = 144 ways
Hence, the number of ways the task can be assigned is 144
Approach Solution 3:
The problem statement informs that:
Given:
- There are 6 tasks T1, T2, T3, T4, T5, T6
- There are 6 people P1, P2, P3, P4, P5, P6.
- Task 1 cannot be assigned either to person 1 or to person 2
- Task 2 must be assigned to either person 3 or person 4.
- Every person is to be assigned one task.
Find out:
- the number of ways the task can be assigned.
Task 2 can only be assigned to two persons i.e. person 3 or person 4.
The number of ways of allocating task 2 = 2 ways.
The first task can be accomplished in 3 ways by 3 persons.
The third task can be accomplished by 4 persons = 4 ways.
Likewise, for the fourth, fifth and sixth tasks, the number of ways is 3, 2 and 1 respectively.
Therefore, total number of ways = 2*3*4*3*2*1 = 144 ways
“There are 6 tasks and 6 persons. Task 1 cannot be assigned either to person 1 or to person 2; Task 2 must be assigned to either person 3 or person 4”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “501 GMAT Questions”. GMAT Problem Solving questions test the candidate’s analytical knowledge in solving mathematical problems. GMAT Quant practice papers consist of various kind of questions that helps the candidates to enhance their mathematical skills.
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