How to simplify or calculate a formula with very big factorials

You need Stirling's approximation. It is very accurate for large factorials.

You don't need to compute the individual factorials in order to compute $k!/(k-i)!i!$, since that's the binomial coefficient $\binom{k}{i}$. A simple algorithm for computing binomial coefficients can be found on Wikipedia. A more sophisticated algorithm is due to Goetgheluck (JSTOR); implementations can be found here and here.

Of course, with numbers of the size that you have, this might still not be feasible, and in this case I also recommend Stirling's formula.

Here is a good writeup about implementing Stirling's approximation, along with a reference implementation:

http://threebrothers.org/brendan/blog/stirlings-approximation-formula-clojure/