Probability that 5 different faces come up twice each if 6 side die is rolled ten times? Find the probability that 5 different faces come up twice each if 6 side die is rolled ten times?
What methods should I apply here?
 A: Hint:
First, how many ways are there to pick 5 of the 6 faces.
If I have chosen 5 different faces, how many different orders can they come in.  What is the probability that they come in that order.
Now multiply everything together.
From this I get:  $$\frac{10!}{2^{5} 6^9}$$
A: Observe that, if five different faces come up twice each in ten rolls, then each of those faces comes up exactly twice and the sixth face doesn't appear at all. Let's see how many ways this can happen.
There are 6 ways to choose which face doesn't come up. For convenience, say "6" is the face that doesn't come up. We now want the rolls 1, 1, 2, 2, 3, 3, 4, 4, 5, and 5, but how many ways can we get these rolls? I'll leave that part to you. (Hint: Use multinomial coefficients.) So, 6 times whatever you get for that part tells you how many acceptable outcomes there are. Divide this by the total number of possible outcomes (6^10).
A: Just for kicks, here is a fun R simulation. 
Running 10 repetitions of a 10,000 simulation run, we get an avg of 0.01178 ± 0.001
#  PARAMETERS:   #
# -------------- #
  set.seed(1)
  sides <- 6
  rolls <- 10

  # simulations to get a probability
  simulations <- 10000

  # repetitions to repeat the process, to get an average result
  reps <- 10


#  ROLL THE DIE  #
# -------------- #
  # This function is a single 10-roll sample
  OneTrial <- function(sides, rolls) { 
    outcome <- sample(sides, rolls, TRUE)

    # count the number of sides that came up exactly-two
    #    and check if that amounts to 5 faces 
    sum(table(outcome) == 2) == 5
  }

# --------------- #
#  MULTIPLE REPS  #
# --------------- #
results <- c()
for (i in 1:reps) {
  simulOutcomes <- replicate(simulations, OneTrial(sides, rolls))
  results[[i]]  <- sum(simulOutcomes) / simulations
}


mean(results)

