Statistics: combinations in Python

Question

I need to compute combinatorials (nCr) in Python but cannot find the function to do that in math, numpy or stat libraries. Something

Answer 1

Here's another alternative. This one was originally written in C++, so it can be backported to C++ for a finite-precision integer (e.g. __int64). The advantage is (1) it involves only integer operations, and (2) it avoids bloating the integer value by doing successive pairs of multiplication and division. I've tested the result with Nas Banov's Pascal triangle, it gets the correct answer:

def choose(n,r):
  """Computes n! / (r! (n-r)!) exactly. Returns a python long int."""
  assert n >= 0
  assert 0 <= r <= n

  c = 1L
  denom = 1
  for (num,denom) in zip(xrange(n,n-r,-1), xrange(1,r+1,1)):
    c = (c * num) // denom
  return c

Rationale: To minimize the # of multiplications and divisions, we rewrite the expression as

    n!      n(n-1)...(n-r+1)
--------- = ----------------
 r!(n-r)!          r!

To avoid multiplication overflow as much as possible, we will evaluate in the following STRICT order, from left to right:

n / 1 * (n-1) / 2 * (n-2) / 3 * ... * (n-r+1) / r

We can show that integer arithmatic operated in this order is exact (i.e. no roundoff error).