If using a 3 sided dice, what is the probability of landing on a 1 exactly 3 times and on a 2 exactly two time?

Question

If using a 3 sided dice and you roll the dice 7 times, what is the probability of landing on a 1 exactly 3 times and on a 2 exactly two times?

Answer 1

By the multinomial distribution, denoting $x_i$ as the number of times that you land on $i$, and where $p = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$, it appears you have:

$$P(x_1 = 3, x_2 = 2, x_3 = 2) = \frac{7!}{3!2!2!} \Big(\frac{1}{3}\Big)^3\Big(\frac{1}{3}\Big)^2 \Big(\frac{1}{3}\Big)^2 = \frac{70}{729}$$