Seven, (Full marks 12 points) There are several table tennis teams, and players from different teams all played one match against each other, while players from the same team did not play against each other. The match statistician recorded the following: A total of 10 players participated in this competition, and a total of 27 matches were played.
(1) How many table tennis teams participated in this competition? Why?
(2) How many players are there in each of these teams?
If a team has at least 7 members, then at least $7 \times 6 \div 2=21$ matches must be reduced, so each team has at most 6 members.

If each team has at most 5 members, then the possible number of members in each team (which will not reduce the number of matches) is as follows: it can be seen that there must be a team with exactly 6 members.
Thus, the number of teams and the corresponding number of members in each team can only be:
(1) $6,3,1$; the reduced number of matches is $15+3=18$;
(2) $6,2,2$; the reduced number of matches is $15+1+1=17$;
(3) $6,2,1,1$; the reduced number of matches is $15+1=16$.

Therefore, only scenario (1), i.e., there are three teams with 6, 3, and 1 members respectively, they have a total of 10 members, and they play exactly $45-18=27$ matches among themselves.