Question 96
To prepare for the last POFM test of the year, Jean-Baptiste and Marie-Odile have collected $100$ mathematics problems and are making a revision schedule. During the $100$ days before the test, each of them must work on one problem per day. Let $x$ be the number of problems Jean-Baptiste works on strictly before Marie-Odile, and let $y$ be the number Marie-Odile works on strictly before Jean-Baptiste. A revision schedule is called fair if $x=y$.
Prove that there are at least $100!\left(2^{50}+(50!)^2\right)$ fair schedules.
Prove that there are at least $100!\left(2^{50}+(50!)^2\right)$ fair schedules.