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

Directly counting the arrangements will be cumbersome. Think why.
Say there are 5 options for first seat (excluding Tom).
Now the number of options for second seat will depend on who takes the first seat. If Jerry does then there are 5 options, but if someone else does, then there are 4 options.
and so on...

So instead, focus on trying to get the unacceptable arrangements.
Total arrangements of 6 people on 6 seats = 6! = 720

Unacceptable arrangements:
Tom sits on the first seat = 5!
Jerry sits on the second seat = 5!
Donald sits on the fourth seat = 5!
But note that there is double counting here - When Tom sits on the first seat, there are 5! ways of arranging others including ways in which Jerry sits on the second seat and/or Donald sits on the fourth seat. These ways are counted again in the next 5!

It becomes a sets question. There is overlap in these sets of 5! each. Think of your venn diagram of three overlapping sets. There are 4! cases in which Tom and Jerry both are sitting on unacceptable seats. Similarly for Donald too. There are 3! cases in which all three are sitting on unacceptable seats.

So Total unacceptable cases = 5! + 5! + 5! - 4! - 4! - 4! + 3! = 3*5! - 3*4! + 3! = 3! * (60 - 12 + 1) = 294

(Using the formula n(Total) = n(A) + n(B) + n(C) - n(A and B) - n(B and C) - n(C and A) + n(A and B and C)

Acceptable cases = 720 - 294 = 426

Answer (A)
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Step 1:
Total possibilities without Tom being seated at 1st position =
5*5*4*3*2*1 = 600

Step 2:
Calculate the possibilities of Jerry sitting at the second position, barring Tom in first "
4*1*4*3*2*1
that is 96

Step 3:
Calculate the possibilities of Donald sitting at 4th, barring Tom being at first
4*4*3*1*2*1 = 96

Step 4:
Now If Jerry being in second & Donald in 4th, Tom not being in first (this is calculated because it is part of both step 2 & step 3):
3*J*3*D*2*1=18
Hence we'll have to add this in the final calculation.

Therefore,
The final possible different arrangements are = Output of (Step 1) - output of (step 2) - output of (Step 3) + output of (step 4)
= 600 - 96 - 96 + 18
= 426

That is option A.

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
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