SOLUTION: How many ways can 6 people sit together in a row a) If two people sit together? b) How many ways can 6 people sit in a circle of two people sit together? Please explain my s

With no restrictions, 6 people can sit in 6! = 720 ways



Having two people sit together means we can treat those two as a unit, and 

we get effectively 5 people to be arranged:  5! = 120 ways 



BUT we must multiply this by 2! = 2 because the two-person unit can be formed in 2! ways (AB and BA):  2*120 = 240.



a)  240 ways







b)  For this part, the physical configuration changes things.  It is like the row  A-B-C-D-E-F but curved around so F is actually understood to be next to A.  

I'll try to draw it:

         

           B - C

         /       \

        A         D

         \       /

          F -  E



This greatly reduces the number of permutations (seating arrangements).  Here we assume it doesn't matter if, say,  F is at "7 O'clock"  --- if we rotated the above arrangement so F was at, say, "11 O'clock" but kept all the relative positions intact, it would still be the same arrangement.

 

The process:

Seat A anywhere.  'A' serves as a reference point in a way, leaving 5 seats for the others to occupy around A.  Those 5 others can be arranged in 5! = 120 ways.   And that is all of the ways they can be seated.  



b) 120 ways



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EDIT:  Oh sorry, I think you wanted the circle configuration for two people sitting together (say AB, BA)...  In this case it is like before: you effectively have 5 people (with one "person" a two-person "unit") so you have 4! = 24 ways to arrange those, times 2! = 2 ways to arrange the two-person unit (AB or BA).  That's 2*24 = 48 ways.



 



 

In part b), the correct  ANSWER  is  2*4! = 2*24 = 48 different possible ways.


Again, we consider these two persons as one unit and place them as a reference unit.


Then we order/permute the remaining  6 - 2 = 4 persons in 4! = 24 ways.


We multiply the number 24 by two, since there are two way to order that two persons, who are sitting together.