Fastest way to generate binomial coefficients

Question

I need to calculate combinations for a number.

What is the fastest way to calculate nCp where n>>p?

I need a fast way to generate binomial coefficients for a

Answer 1

The fastest reasonable approximation in my own benchmarking is the approximation used by the Apache Commons Maths library: http://commons.apache.org/proper/commons-math/apidocs/org/apache/commons/math3/special/Gamma.html#logGamma(double)

My colleagues and I tried to see if we could beat it, while using exact calculations rather than approximates. All approaches failed miserably (many orders slower) except one, which was 2-3 times slower. The best performing approach uses https://math.stackexchange.com/a/202559/123948, here is the code (in Scala):

var i: Int = 0
var binCoeff: Double = 1
while (i < k) {
  binCoeff *= (n - i) / (k - i).toDouble
  i += 1
}
binCoeff

The really bad approaches where various attempts at implementing Pascal's Triangle using tail recursion.