# sample space of picking balls

A bag contains red and black balls. We choose 3 balls at random. What is the sample space of this experiment?

We $3! = 6$ ways of choosing 3 balls.

But when listing them I get 8 ways:

$S = \{RRR, RRB, RBR, RBB, BBB, BRB, BRR, RRB\}$

You have $3!$ ways of ordering three different balls you have already drawn (first ball has three places to choose from etc.): $$\{ABC, ACB, CAB, CBA, BCA, BAC\}$$ You have $2^3$ different color orders you can make when you draw balls from a bag (first ball can be one of two different colours etc.): $$\{RRR, RRB, RBR, RBB, BRR, BRB, BBR, BBB\}$$ Correctly identifying what method to use to count exactly what you want to count (and not something else) is difficult and more or less only comes with experience. Keep up the work, and hopefully it sticks sooner or later.