A team made up of $3$ boys and $3$ girls is to be selected from $12$ boys and $9$ girls for a competition. Tom and Henry are two of the boys in the group. Determine the number of ways the team may be chosen if Tom and Henry are either both included or both excluded.
A cinema’s back row contains $12$ seats, and all of them are empty. A party of $8$ people, including Mary and Frances, are seated in this row. Determine the number of distinct ways they can be seated in these $12$ seats when there are no restrictions.
Find the number of different ways they can be seated if Mary and Frances are not in seats next to one another.
Determine the number of different ways they can be seated if all $8$ people are together, with no empty seats between them.