Exhibition game
However, most of the major FBS teams annually schedule early season non-conference preseason home games against lesser opponents that are lower-tier FBS, Football Championship, or Division II schools, which often result in lopsided victories in favor of the FBS teams and act as exhibition games in all but name, though they additionally provide a large appearance fee and at least one guaranteed television appearance for the smaller school. These games also receive the same criticism as NFL exhibition games, but instead it is targeted to schools scheduling low-quality opponents and the simplicity for a team to run up the score against a weak opponent. However, these games are susceptible to backfiring, resulting in damage in poll position and public perception, especially if the higher ranked team loses, although the mere act of scheduling a weak opponent is harmful to a team's overall strength of schedule itself. Games an FBS team schedules against lower division opponents do not count toward the minimum seven wins required for bowl eligibility, and only one game against an FCS team can be counted. With the start of the College Football Playoff system for the 2014 season, major teams are now discouraged from scheduling weaker opponents for their non-conference schedule because of a much higher emphasis on strength of schedule than in the Bowl Championship Series era.

Q: How many games between FBS and FCS teams count towards the FBS's bowl eligibility?
one