This isn't completely clear to me but I read it to say, for example - 100 participants. one called X - 7 tournaments T1 to T7. Say 8 participants per tournament.

T1 pick 8 participants at random from 100, T2 pick 8 from 99, then T3 ... to T7 pick 8 from 94.

What is the probability that X has been in one of the tournaments?

Probability of X NOT being picked is 92/100x91/99x...x86/94 and the probability of being picked as 1 - that. This can all be condensed into 1-92!/85!x93!/100!

Putting this into Excel terms, probability of being picked is =1-FACT(p-n)/FACT(p-n-t)*FACT(p-t)/FACT(p) where n is the number in each tournament. There is no formula to solve this for n, but n is easily found using a spreadsheet for any given p and t.