When
Since we are adding up to 11!, can this be done without individual?
You might be able to use Wilson's Theorem:
if and only if is prime.
In this case, you have . You can use this fact to get by multiplying by on both sides. You can get by multiplying by after that, and etc...
This method may not really save much time though - you still need to compute modular inverses and peel off terms in the factorial, one by one. Is it really so bad to compute the factorials by hand?