There are 6 tasks T1, T2, T3, T4, T5, T6
and 6 people P1, P2, P3, P4, P5, P6.

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, P6.
T3 could be assigned to each of 4 remaining people.
T4 could be assigned to each of 3 remaining people.
T5 could be assigned to each of 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 answer is 2*3*4*3*2*1=144

The answer is A.
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Take the task of assigning the tasks and break it into stages.

Start with the most restrictive stage(s)!

Stage 1: Select a person to complete task 2
Since task 2 must be assigned to either person 3 or person 4, 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 assigned a task
However, task 1 cannot be assigned to person 1 or to person 2
So, there are only 3 people to choose from.
So we can complete stage 2 in 3 ways

Stage 3: Select a person to complete task 3
There are 4 people remaining, so we can complete stage 3 in 4 ways

Stage 4: Select a person to complete task 4
There are 3 people remaining, so we can complete stage 4 in 3 ways

Stage 5: Select a person to complete task 5
There are 2 people remaining, so we can complete stage 5 in 2 ways

Stage 6: Select a person to complete task 6
There is 1 person remaining, so we can complete stage 6 in 1 way

By the Fundamental Counting Principle (FCP), we can complete all 6 stages (and thus assign all 6 tasks) in (2)(3)(4)(3)(2)(1) ways (= 144 ways)

Answer:

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

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Let T1,T2,T3,T4,T5 &T6 be the Tasks and P1,P2,P3,P4,P5 & P6 be the Persons

Task T2 can be done only in 2 ways (either to P3 or P4)

Task T1 can be done in 3 ways (P5,P6, and 1 among P3/P4)

Task T3 can be done in 4 ways (Remaining 4 people)

Task T4 can be done in 3 ways (Remaining 3 people)

Task T5 can be done in 2 ways (Remaining 2 people)

Task T6 can be done in 1 way (Remaining 1 person)

So, totally there can be a combination of 2 * 3 * 4 * 3 * 2 * 1 ways = 144 Ways.

Hence option A is the answer

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?

(1) 144

(2) 180

(3) 192

(4) 360

(5) 716
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Task 1 to person 3 > Task 2 to person 4:
1*1*4*3*2*1 = 24

Task 1 to person 4 > Task 2 to person 3:
1*1*4*3*2*1 = 24

Task 1 to person 5 or 6:
2*2*4*3*2*1 = 96


Total: 24+24+96 = 144

Merging topics. Please check the discussion above.
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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?

Combination # 1:
T1 -> P3 , T2 -> p4 then T3 will have 4 options, T4 3 options, T5 - 2 options and T6 - 1 option
1*1*4*3*2*1 = 24

Combination # 2:
T1 -> P4 , T2->P3, then others will have same options as above
1*1*4*3*2*1 = 24

Combination # 3:
T1 is assigned to person 5 or person 6.
T2 can be assigned to person 3 or person 4.
2*2*4*3*2*1 = 96

Adding all together = 24+24+96 = 144

Ans: A
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