We roll five six-sided fair dice. a) What is the probability that we get a three dice of one face and two of a second face? (For example, an outcome like 55533.) Explain your counting process. b) What is the probability that the dice all show different faces? Explain your counting process.
a) I worked the problem, but it felt so simple that I don't trust it. For $5$ rolls, you have $6$ options, so the number of possible outcomes is $6^5$. For the first roll, you have $6C1$ choices, for the second and third roll, $1C1$ choice. For the fourth roll you have $5C1$ choices, and then $1C1$ for the fifth roll, Making the probability $(6*5)/6^5$?
b) This one I'm more confident in. Number of possible out comes is still $6^5$, with us having one less option with each roll. $(6*5*4*3*2)/6^5$