A lot more than half of Europe descends from Charlemagne. It is closer to the reality to say that literally all people living today who are of modern European extraction (inclusive of people of European descent living in the parts of Africa, the Americas, Asia, Australia, and Oceania that were colonized by European powers or by the United States) are descended from Charlemagne. Three factors influence this: First, Charlemagne fathered 18 children. Second, 1,205 years have elapsed since Charlemagne died, which, if we aproximate a new generation for every 20 years, means that 60 generations have since been born. Third, the Black Death killed somewhere between 30% and 60% of the population of Europe in the span of four years during the mid-14th Century, which created a major population bottleneck among Europeans that homogenized the European gene pool to a nigh incomprehenisble degree. All of these three factors combined make it essentially impossible for anyone of European extraction today to not be a descendant of Charlemagne; in fact, the latter two factors make it unlikely that anyone of European descent living today is not a descendant of, basically, every person who lived in Europe in the 9th Century and has descendants who are living today.
