So we have 5 persons around each table.


The key fact in the solution is that if there are N people, then there are (N-1)! distinguishable ways 
to place them around the round table.



    These are so called "circular" permutations, in distinct from usual permutations, 

    where  N!  regular permutations exist for N objects.



After this introduction, I am ready now to present the solution.


1)  You can select 5 persons from 10 to place them around first table (team 1) by   =  = 252 ways.


    As soon as your team 1 is just formed, team 2 (for the second table) is formed automatically - they are remaining 5 people.



2)  You can place 5 members of the team 1 around Table 1 by 4! = 24 ways.



3)  You can place 5 members of the team 2 around Table 2 by 4! = 24 ways.



4)  Thus the total number of different ways is the product


         * 4! * 4! = 252 * 24 * 24.


    Use your calculator to get the required number.