John draws a regular five pointed star in the sand, and at each of the 5 outward-pointing points and 5 inward-pointing points he places one of ten different sea shells. How many ways can he place the shells, if reflections and rotations of an arrangement are considered equivalent?

Any help is greatly appreciated

Memes4Life132
Jul 21, 2018

#1**+1 **

Diregarding my previous answer - I didn't consider that all rotations were the same - I think this is like the total number of arrangements of 10 people seated at a table...

We place "anchor" any shell in any position.....and we have 9! ways to place the other shells

So...the total arrangements are 9! = 362880

CPhill
Jul 21, 2018

#3**+1 **

Note that we can "anchor" any one of the shells at any of the specified positions....at the next position [ let's assume that we are moving "clockwise" around the star...but..it doesn't matter, we could also move "counter-clockwise" with the same results ], we can choose any 1 of the other 9 shells = 9 possibilities

Similarly, at the next position, we could choose any 1 of the 8 remaining shells = 8 possibilities

So...continuing this pattern at each successive position we get

9 * 8 * 7 * 6 * 5 *4*3* 2* 1 = 9! = 362880 arrangements

CPhill
Jul 21, 2018