Greg, Marcia, Peter, Jan, Bobby and Cindy go to a movie and

@krishp84


We cannot apply symmetry because the number of cases where M and J are sitting together is not equal to the number of cases where M and J are not sitting together.
Consider 3 people: A, B and C
They can be arranged in 3! = 6 ways
ABC
ACB
BAC
BCA
CAB
CBA
In how many of these are A and B sitting together? 2
In how many are they not sitting together? 4
When we arrange 6 people here in 6! (= 720) ways, in how many of those will M and J sit together?
Consider M and J to be one and arrange 5 people in 5! ways. Also, M and J can exchange places so multiply by 2.
You get 5!*2 = 240 ways
Out of 720, M and J will be next to each other in only 240 ways.



That question would be something like this: 6 people go to a movie and sit next to each other in 6 adjacent seats in the front row of the theatre. If Marcia will not sit to the right of Jan, how many different arrangements are possible?

'to the right of Jan' means anywhere on the right, not necessarily on the adjacent seat. Here we see symmetry because there are only 2 ways in which Marcia can sit. She can sit either to the left of Jan (any seat on the left) or to the right of Jan (any seat on the right). There is nothing else possible. The number of cases in which she will sit to the left will be same as the number of cases in which she will sit to the right. That is why the answer here will be 6!/2.

But the original question talks about sitting right next to each other on adjacent seats. The probability of sitting 'next to each other' is less than the probability of sitting 'not next to each other'. Hence we cannot apply the symmetry principle there.