Just 2 years ahead

Considering $n! = (n-1)! n$ the reminder of each term in summation $r_k=k! \pmod{2017}$ can be expressed from previous reminder as $r_k = r_{k-1} k \pmod{2017}$ . The result is then $\displaystyle\sum_{k=1}^{20} r_k \pmod{2017}$ .

It can be easily evaluated in python 'Coding environment' using first or second code sample:

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M=2017
N=20

s,r = 0,1
for k in range(1,N+1):
    r = (r*k)%M
    s += r

print(s % M)

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# For one-liners and pure functionalists
from itertools import accumulate
print(sum(accumulate(range(1,20+1),lambda r,k: (r*k)%2017))%2017)