Dinner plate combinatorics problemSolutionTo set 6 plates of

To set 6 plates of aa patterns in a line =a6=a6

To set 7 plates of aa patterns in a circle =a7a7+a=a7a7+a How did we get that. We set 7 plates in a line, a7a7Either the plates are all one color (aa possibilities) or there is at least one plate that is different from the rest. If there is one plate different from the rest, then we need to account for the circularity of the table.

Since 7 is prime, this is the only special consideration.

6 people at the table special cases.

Every person across the table has the same plate. (2-fold symmetery) every other person has the same plate. (3-fold symmetry) everyone has the same plate. (6-fold symmetry)

Every person across the table has the same plate. arrange 3 plates, on one side of the table, and the other side is fixed. We want to exclude all the same, as we are counting it in its own catagory. a2(a1)=a3aa2(a1)=a3a

Every other person has the same plate. a2aa2a

Asymmetric arrangements: a6(a3a)(a2a)aa6(a3a)(a2a)a

a6(a3a)(a2a)a6+a3a3+a2a2+a