G, M, P, J, B and C are supposed to sit in 6 different seats. But M an

Question

G, M, P, J, B and C are supposed to sit in 6 different seats. But M and J won't sit together, how many different arrangements are possible?

OA:  480

So, I know I can calculate the possibilities w/o constraints and subtract the possibilities with constraints and get the answer as 6! - 5! to get 480. What I am trying to think is in terms of the following (which works for most problems, but I can't figure out why this model doesn't fit this situation)

I have 6 possibilities for Seat 1 and I assume its M, so possibilities = 6.
Now, I have 4 possibilities for Seat 2 because J won't sit with M, so possibilities = 4
Now, I have 4 possibilities again for Seat 3 because M is back in the group, so possibilities = 4
And then finally 3 and 2, so possibilities = 3 * 2

Multiplying, 6 * 4 * 4 * 6 = 576 possibilities.

Where am I formulating this wrongly?

Thanks in advance.

firasath · Answer

To solve this you can do one of two things:
1. find the total number of outcomes and then subtract the number of invalid outcomes based on the constraints
2. Count the number of valid outcomes based on the constraints

1) total number of outcomes = 6!
 invalid outcomes would be when M & J sit together can be visualized by the following:
 
M J _ _ _ _ 
_ M J _ _ _ 
_ _ M J _ _
_ _ _ M J _
_ _ _ _ M J

this equals 5*4! = 5!
but you need to consider that M J do not need to be in that order, it could be J M, so multiply your outcomes by 2. Therefore your total invalid outcomes based on the constrains is 2*5! = 240

and thus the answer is 6! - 2*5! = 720-240 = 480

(looks like you forgot the 2 multiplier in your answer)

2) the number of valid outcomes can be visualized as follows:

M _ J _ _ _ 
M _ _ J _ _ 
M _ _ _ J _ 
M _ _ _ _ J

_ M _ J _ _ 
_ M _ _ J _
_ M _ _ _ J

_ _ M _ J _
_ _ M _ _ J

_ _ _ M _ J

so we have (4+3+2+1)*4! = 10*4!

but we have to consider that M J do not to need to be in that order, it could be J M, so double what we got to get: 2*10*4! = 480

your logic in trying to this problem using approach #2 is off. Although you can use this method when doing permutations, the constraints don't allow you to do it this way because depending on who is in the first seat the values for the other seats will be different. Avoid using this when there are dependencies on position.
In your calculations, first you are saying that there is 6 possibilities for the first seat, but then you calculate for when that seat is for M, this would invalidate the first 6 because setting that seat to M as in your assumption would make the first value a 1.

whenever you see dependencies, keep it simple and avoid confusion, use the following principles:
1. find the total number of outcomes and then subtract the number of invalid outcomes based on the constraints
2. Count the number of valid outcomes based on the constraints

Hope that helps..

G, M, P, J, B and C are supposed to sit in 6 different seats. But M an

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