# Coloring of the edge of the 1*3 grid

I am trying to find the number of distinct coloring in the following problem:

Consider a 1*3 grid (shown as below) using 10 sticks and 8 balls. Color sticks with $$m$$ colors. How many ways are there to color the sticks with $$m$$ colors?

I try to use Burnside's lemma to count the distinct coloring, but I cannot find all the symmetric groups. Any help would be appreciated.

• This looks to have only four automorphisms which are easily factored into cycles. – Marko Riedel Nov 12 '18 at 14:30
• Thank you for your comment. I think the symmetries are identity, 180 rotations, reflection with respect to the horizontal axis of symmetry and vertical axis of symmetry. Am I right?@MarkoRiedel – Morteza Soltani Nov 12 '18 at 19:20
• That would be correct. These can be factored from the table notation or more effectively, by inspection. – Marko Riedel Nov 12 '18 at 21:25
• @MarkoRiedel Thank you for your response. I have found all the symmetries of this figure and have solved my problem. – Morteza Soltani Nov 13 '18 at 1:04