There are 120 different possible seating arrangments. 

Note that on a round table ABCDEF and BCDEFA is the same. 

The first person can sit on any one of the seats. Now, for
the second person there are 5 options, for the third person
there are 4 options, for the forth person there are 3
options, for the fifth person there are 2 options and for
the last person there is just one option. 

Thus, total different possible seating arrangements are 

= 5 * 4 * 3 * 2 * 1 

= 120