Problem:
Every positive integer k has a unique factorial base expansion (f1​,f2​,f3​,…,fm​), meaning that k=1!⋅f1​+2!⋅f2​+3!⋅f3​+⋯+m!⋅fm​, where each fi​ is an integer, 0≤fi​≤i, and 0<fm​. Given that (f1​,f2​,f3​,…,fj​) is the factorial base expansion of 16!−32!+48!−64!+⋯+1968!−1984!+2000!, find the value of f1​−f2​+f3​−f4​+⋯+(−1)j+1fj​.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions