- How many round necklace can be made using $10$ different color beads?
- If $10$ members are to be seated on a round table then how many arrangements are possible?
Answer:
First problem has got answer $=9!/2$ and second $=9!$
why in first we need to divide by $2$?
Look at the axes of symmetry. The necklace and desk are symmetric to rotation of 36 degrees. This means we need to divide the number of permutations (10!) by 10. (360/36)
The necklace is further symmetric to rotation of 180 degrees around an axis passing through it. This means we need to divide the number of permutations (10!) by 10 and then by 2
Why are we dividing - let's say we order the beads as yellow, green,..., red and then while you're not looking we rotate the chain so we now see it as red, ...., green, yellow. You can't tell if we originally had red, ...., green, yellow, or if we rotated it. Therefore the two arrangements are equivalent.