SOLUTION: A bag contains three black balls, four white balls, and five red balls. Three balls are removed without replacement. What is the probability of of obtaining at least two red?

First we calculate the number of ways of removing any 3 balls from the 3+4+5=12
balls. This is 12C3 = 220 ways to remove any 3 balls.

We first find the number of ways we don't want, which is the number of obtaining
either no reds or just 1 red.

The number of ways of getting no reds is the number of ways of choosing 3 from
the 3+4=7 non-red balls.  This is 7C3 = 35 ways.

The number of ways of getting exactly 1 red is the number of ways of: 

1. choosing 2 from the 3+4=7 non-red balls.  This is 7C2 = 21 ways.
and for each of these 21 ways,
2. choosing the 1 red ball. This is 5C1 = 5 ways
That's 21∙5 = 105 ways to get exactly 1 red ball.
  
So the total number of ways to get what we don't want is 35+105 = 140 ways.

That leaves 220-140=80 ways we want.

So the desired probability is 80 ways out of 220 or 80/220 which reduces to

4/11.

Edwin