Probability of 1 dice rolled N times

Question

In this scenario, N = 3. The dice has 6 faces.

I want to know what is the probability of rolling a 4,5,6 in only a single dice when rolled three times regardless of order.

I understand that [4,5,6]/6 is 3/6 where in there is 50% chance of rolling it. I also understand that there is a 12.5% chance of rolling 4,5,6 in each dice when rolled three times. (correct me if I'm wrong here)

Answer 1

I think that the question is clear enough, even if OP's initial attempt at an answer is not productive. (Joined by authors of several unhelpful comments.)

The probability of getting 456 in exactly that order is $1/6^3$, similarly for getting 546, and so on. So the answer is $3!/6^3 = 1/36.$

For doubters, here is a simulation, in R statistical software, of a million such experiments, which should be accurate to about two or three places. (There are more efficient ways to program this, but maybe the one shown is among the easiest to follow.)

m = 10^6; n = 3; yes=logical(m)
for (i in 1:m) {
   faces = sample(1:6, n, repl=T)
   f4 = sum(faces==4); f5 = sum(faces==5);  f6=sum(faces==6)
   yes[i] = (f4==1 & f5==1 & f6==1) }
mean(yes);  1/36
## 0.027463     # aprx prob of event
## 0.02777778   # exact probability