binomial coefficient (by hand)

Does anybody have or know where to get the code for computing the

binomial coefficient as you'd do it on a piece of paper/manually?

The problem I having are due to approximations made my the factorial

routine as described by Numerical Recipes. The error increases when

further approximations are made when I try to compute the binomial

coefficient and then the hypergeometric distribution.

If sumeone is familiar with hypergeometric distribution, to obtain the

p-value, you take the -ln of the sum of probabilities and subtract

them from 1. Well, the problem is that due to the approximations

described above, when I sum all probabilities the total is greater

than one by 1.0e-6; and probability cannot be defined as negative. In

reality the correct answer should be something very close to 1.0.

How does one encounter these numerical errors when trying to prodict

probabilities of such a low chance?