Tom, Jerry, and Donald and other three people sit in a line. From left

I am unable to understand the solution

According to me its coming like,
5x4x4x2x2x1= 320 (which is none of the options)

starting from left to right as there are 5 ways of filling the first place, 4 ways of filling the second place, 4 ways for the third place 2 ways for the fourth place 2 for the fifth and one for the final place.

Can someone tell me where am I getting the concept wrong?

Goal: going to calculate the total arrangements and then subtract out the unfavorable outcomes.

Also going to use overlapping sets theory to determine the correct amount of arrangements to remove as unfavorable outcomes.


(1st) 6! = 720 arrangements with no conditions

(2nd)finding the correct number of arrangements to remove:


Case 1: # of arrangements with T in spot 1

1 * 5 * 4 * 3 * 2 * 1 = 5!

Case 2: # of arrangements with J in spot 2

5 * 1 * 4 * 3 * 2 * 1 = 5!

Case 3: # of arrangements with D in spot 4

Same 5!


We have over counted the arrangements.

In case 1, we have included the times when

-just T violates his condition

-T and J violate their conditions

-and T and J and D all violate their condition


Following the set theory formula of:
(A + B + C) - ( A&B + A&C + B&C) + (A & B & C)

(1st) we are going to subtract out the arrangements for each time that 2 of the 3 people violate their condition

And

(2nd)going to add back the arrangements for each time all 3 violated the condition


T is in 1st spot and J is in 2nd spot: 1 * 1 * 4 * 3 * 2 * 1 = 4!

T is in the 1st spot and D is in the 4th spot —- 4!

J is in the 2nd spot and D is in the 4th spot —— 4!


Lastly, all 3 violate their condition:
T in 1st spot and J in 2nd spot and D in 4th spot = 1 * 1 * 3 * 1 * 2 * 1 = 3!


Total Number of Arrangements in which the conditions are violated (I.e., unfavorable outcomes) =

(5! + 5! + 5!) - (4! + 4! + 4!) + (3!) =

(360) - (72) + (6) =

294 arrangements we want removed with no over counting


720 arrangements with no condition

-

294
————-

426 ways for the people to sit such that T is not in 1st seat, J is not in 2nd seat, and D is not in 3rd seat

(A)

Posted from my mobile device

If you are finding it difficult to understand the question, I am breaking it down for you.
The below solution is broken down to its basic constituents. Obviously it's a longer approach.
After looking at this, you may check the above solutions and hopefully you would understand better.



Answer A

PS. Attached an image since it was too much to write

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.