Project Euler Problem 365
The binomial coefficient displaystyle{binom{10^{18}}{10^9}} is a number with more than 9 billion (9times 10^9) digits.
Solution
Answer: 162619462356610313
After reducing the computation using Lucas’ theorem modulo each prime $p\in(1000,5000)$, evaluating
$$\binom{10^{18}}{10^9}\bmod p$$
for every prime $p$, and then combining residues for each triple $(p,q,r)$ using the Chinese Remainder Theorem over $pqr$, the required sum over all primes $1000<p<q<r<5000$ is:
Answer: 162619462356610313