SOLUTION: In how many ways can 6 people be arranged to sit around a circular table with 10 chairs?

Working with circular placing/arrangements, we can think that the chairs are numbered from 1 to 10 inclusive, 
and that the chair number 1 is vacant and is placed in position "North" or {12 o'clock".


Then we have 9 chairs from 2 to 10 to place there 6 people.


    1st person can be placed to any of 9 chairs            (9 options);

    2nd person can be placed to any of 8 remaining chairs  (8 options);

    3rd person can be placed to any of 7 remaining chairs  (7 options);

    4th person can be placed to any of 6 remaining chairs  (6 options);

    5th person can be placed to any of 5 remaining chairs  (5 options);

    6th person can be placed to any of 4 remaining chairs  (4 options).


Thus the total number of different placements/arrangements is this product

    9*8*7*6*5*4 = 60480.    ANSWER


It is the product of 6 consecutive integers in decreasing order, starting from 9.