A dictionary gives me these 5 possible treatments of the term "spinner".


    1 - one that spins

    2 - a fisherman's lure consisting of a spoon, blade, or set of wings that revolves when drawn through the water

    3 - a conical sheet metal fairing that is attached to an airplane propeller boss and revolves with it

    4 - a movable arrow that is spun on its dial to indicate the number or kind of moves a player may make in a board game

    5 - spin doctor.


Of these possible treatments, I will use #4  as the most adequate to the problem.


Then the adequate mathematical reformulation of the problem is THIS:

    How many circular permutations are possible for 10 different numbers ?


The answer is commonly/widely known:

    The number of different circular permutations of 10 different items is  (10-1)! = 9! = 9*8*7*6*5*4*3*2 = 362880.