There are two distinct round tables, each with 5 seats. In how many ways may a group of 10 be seated?
There's a few ways to approach this - let me show you one.
Let's first notice that when dealing with seating at tables, we'll have to work with the lack of definite seat numbers (as opposed to a row of seats that has definite end seats). So what I want to do is break down the problem into the numbers of ways I can have 10 people sit at the two tables, then deal with actual seating.
Putting people at the tables
If we take Table 1 and choose 5 people to be at it, we'll naturally deal with Table 2 (if the people aren't at Table 1, they're at Table 2).
So let's take 10 choose 5:
One thing to notice - when I have persons 1, 2, 3, 4, 5 at Table 1 and 6, 7, 8, 9, 10 at Table 2, it's the same as having persons 6, 7, 8, 9, 10 at Table 1. So to get rid of the duplicates, let's divide by 2:
Seating at the tables
For each table, we have 5 people sitting. If we were dealing with rows of chairs, we'd have
There are two tables, so we multiply by 2:
Putting it together
There are 126 ways to divide the people up into distinct groups at the tables and 48 ways to seat people per group, giving: