Calculate odds of winning (UK) Lottery jackpot
A quick note to start:

Console.WriteLine(String.Format("the odds are 1/{0:n0}", totalWinOdds));


Is exactly the same as:

Console.WriteLine("the odds are 1/{0:n0}", totalWinOdds);


Now on to some maths fun... There's a multiplicative version which means you don't have to compute such massive numbers so don't need to use BigIntegers:

private int GetBinomialCoefficient(int totalNumberOfBalls, int numberOfBallsDrawn)
{
    // range checking of arguments omitted.
    var total = 1;
    for (var i = 1; i &lt;= numberOfBallsDrawn; i++)
    {
        total *= (totalNumberOfBalls + 1 - i) / i;
    }
    return total;
}



  GetBinomialCoefficient(59, 6) == 45057474


Edit

I have changed the name of the method away from GetJackpotOdds based on 200_success's excellent answer.