Finding GCD of a binomial coefficient and a large integer

[smslca]

can we find the value of gcd(x c y , z) easily and very fast using a computer.

where
1. "c" represents "combinations" used in 'permutations and combinations'.
2. x is very very large number (ex: may be of 100 or 1000 numerical digits)
3. y is also large having 2 to 5 digits less than x.
4. z is also large having the same number of digits as x.

 

[Robert1986]

What are some ideas that you have?

Not having put much thought into this, I would try to use the formula for C(x,y) to try to determine the prime factorization.