Well, just as with Problem 16, Mathematica to the rescue!
In[6]:= 100!
Out[6]= 93326215443944152681699238856266700490715968264381621468592963\
8952175999932299156089414639761565182862536979208272237582511852109168\
64000000000000000000000000
In[10]:= list = {9, 3, 3, 2, 6, 2, 1, 5, 4, 4, 3, 9, 4, 4, 1, 5, 2, 6,
8, 1, 6, 9, 9, 2, 3, 8, 8, 5, 6, 2, 6, 6, 7, 0, 0, 4, 9, 0, 7, 1,
5, 9, 6, 8, 2, 6, 4, 3, 8, 1, 6, 2, 1, 4, 6, 8, 5, 9, 2, 9, 6, 3,
8, 9, 5, 2, 1, 7, 5, 9, 9, 9, 9, 3, 2, 2, 9, 9, 1, 5, 6, 0, 8, 9,
4, 1, 4, 6, 3, 9, 7, 6, 1, 5, 6, 5, 1, 8, 2, 8, 6, 2, 5, 3, 6, 9,
7, 9, 2, 0, 8, 2, 7, 2, 2, 3, 7, 5, 8, 2, 5, 1, 1, 8, 5, 2, 1, 0,
9, 1, 6, 8, 6, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0};
Total[list]
Out[11]=