Use The Gamma Function, Use The Symmetry Of The Binomial Coefficient, Use The Taylor Sum, But Please, For The Love Of Euler, Don't Use This Argument For Why 0! = 1

Mathematicians have legitimate, beautiful reasons for why 0! equals 1—the gamma function extension, combinatorial interpretations, the empty product convention—but then someone walks into your office and says "well obviously 0! = 1 because 1! = 1 × 0!" and you can literally feel your brain melting out of your ears. This circular reasoning is the mathematical equivalent of saying "I'm right because I said so." It's like proving water is wet by saying wet things are watery. The argument assumes what it's trying to prove, which is why it causes a full-body headache that makes migraines look like a gentle massage. Fun fact: The real reason 0! = 1 is that there's exactly ONE way to arrange zero objects—by doing nothing. It's the empty arrangement, and it counts. Much more elegant than this recursive nightmare that haunts every math professor's dreams.