There are 4 tasks to be allotted to students. Each task must be

Question

There are 4 tasks to be allotted to students. Each task must be allotted to exactly one of the 7 students: Albus, Betty, Cindy, Dudley, Emma, Fred, and Harry. If Harry doesn’t want to do Task 1 or Task 4 and Dudley does not want to do Task 2 or Task 3, in how many ways can we assign the task?

(A) 100

(B) 200

(C) 240

(D) 280

(E) 440

(A) 100

(B) 200

(C) 240

(D) 280

(E) 440

gmatophobia · Accepted Answer

To assign 4 tasks, we need four out of seven students. Therefore to complete the problem at hand, we need to perform the following two steps -

Select a group of four students from seven students 
  Assign tasks to each student within that group

We can create the group in four different ways -

Group 1 : Harry is a part of the selected four but Dudley is not
 
  Group 2 : Dudley is a part of the selected four but Harry is not
 
  Group 3 : Harry and Dudley both are part of the selected group
 
  Group 4 : Neither Harry nor Dudley is a part of the selected group

Let's calculate the above two steps for each group

Group 1  : Harry is a part of the selected four but Dudley is not

H _ _ _

Step 1 : Select three students from available five students

We can select three students from the five available students in  ^5C_3  ways

Step 2 : Assign the tasks to each student

Harry can be assigned tasks in two ways (either task T2 or T3)  
 The remaining three students can be assigned tasks in 3! ways.

Total =  ^5C_3 * 2 * 3! = 120

Group 2  : Dudley is a part of the selected four but Harry is not

D _ _ _

Step 1 : Select three students from available five students

We can select three students from the five available students in  ^5C_3  ways

Step 2 : Assign the tasks to each student

Dudley can be assigned tasks in two ways (either task T1 or T4)  
 The remaining three students can be assigned tasks in 3! ways.

Total =  ^5C_3 * 2 * 3! = 120

Group 3  : Harry and Dudley both are part of the selected group

D H _ _

Step 2 : Select two students from available five students

We can select two students from the five available students in  ^5C_2  ways

Step 2 : Assign the tasks to each student

Harry can be assigned tasks in two ways (either task T2 or T3)  
 Dudley can be assigned tasks in two ways (either task T1 or T4)  
 The remaining two students can be assigned tasks in 2! ways.

Total =  ^5C_2 * 2* 2 * 2! = 80

Group 4  : Neither Harry nor Dudley is a part of the selected group
_ _ _ _

Step 2 : Select four students from available five students

We can select two students from the five available students in  ^5C_4  ways

Step 2 : Assign the tasks to each student

The tasks can be assigned without any restriction i.e. in 4! ways

Total =   ^5C_4 * 4!  = 120

Number of ways in which we can assign the task = 120 + 120 + 80 +120 = 440

Option E

ThatDudeKnows · Answer

Quick prequel: a clarification is necessary on this question. As written, there is no stipulation that each task is done by a different student. Without that stipulation, the solution should be as follows.
We have 6 choices for who will do Task 1, 6 for Task 2, 6 for Task 3, and 6 for Task 4. That's 6^4 = 1296. Since that's not an option, let's assume that stipulation is in play and the question should be reworded.

Okay, how about a MUCH faster way? 
(Sure the nPr/nCr stuff is great, but a little logic saves a lot of work...and GMAC doesn't care HOW you get the right answer, only THAT you get the right answer.)

Task 1 can be assigned to 6 people. 
Once we've done that, we have 6 people left. If they include Dudley, then we have 5 options for Task 2. If they don't include Dudley, then we have 6 options for Task 2. So, either 5 or 6 options.
Once we've done that, we have 5 people left. If they include Dudley, then we have 4 options for Task 3. If they don't include Dudley, then we have 5 options for Task 3. So, either 4 or 5 options.
Once we've done that, we have 4 people left. If they include Harry, then we have 3 options for Task 4. If they don't include Harry, then we have 4 options for Task 4. So, either 3 or 4 options.

We know that the number of total options CAN'T be smaller than each of the smallest numbers from each row multiplied together. That's 6*5*4*3 = 360. Eliminate A, B, C, and D.

Answer choice E.

Regor60 · Answer

With no restrictions the tasks can be allocated:

7*6*5*4 = 840

Each has two tasks that can't be allocated to them, so the above needs to be reduced, with a preliminary approach as:

2*(3 remaining tasks among 6 people) =

2*6*5*4 = 240

Since this restriction applies to both Dudley and Harry:

2*240 = 480

Subtracting these restrictions from the initial 840:

840-480 = 360

Since we know there is overlap in the restrictions, some number would be added back.

The only answer that fits is:

440

Another way:

H & D each can "participate" or not.

Each can perform two independent tasks, so there are:

4 ways to allocate 2 tasks between them.

This leaves 2 tasks for the remaining 5 people:

5*4 ways

Total ways when both H and D participate:

4*5*4= 80 ways

Now, H can participate and D not, and visa versa, so 2 ways and each has their 2 individual ways:

2*2 = 4 ways

This leaves 3 tasks for the other 5 people or:

5*4*3 = 60

Total ways where one participates and the other doesn't:

4*60 = 240

Finally, neither can participate, in which case 4 tasks must be allocated among 5 people:

5*4*3*2 = 120 ways

Grand total ways:

80+240+120 = 440

Posted from my mobile device

Rutvik01 · Answer

Can it be solved faster?

bumpbot · Answer

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