The techno nerds are arguing about the amount of episodes being either 72 or 73, but www.startrek.com actually lists 80 episodes, spread out over three seasons. Season one comprises 30 episodes, season two has 26, and season three has 24. The two-part episode "The Menagerie" is listed as two seperate episodes.
Another possible point of discussion is that the number 80 also includes the never aired pilot "The Cage", of which bits and pieces were reused in "The Menagerie". So when "The Cage" is not considered to be part of the series, Star Trek has 79 episodes.