Problem: A fair six-sided die is rolled 30 independent times. Let X be the number of ones and Y the number of twos. Find the joint PMF and marginal PMF for each.
My Question: does this problem have joint trinomial distribution and binomial marginal distribution? And Why?
Sample consists of 30-element sequences, each element is either 1, 2, 3, 4, 5 or 6. Marginal PMFs will be the same in both cases. Let $x$ be the number of ones. There ${30 \choose x}$ ways to select positions for ones. The PMF, therefore, is $${30 \choose x}(\frac{1}{6})^x(\frac{5}{6})^{30 - x}$$ which is binomial distribution with $n = 30$ and $p = 1/6$. Perhaps a more intuitive way of explaining is like this: binomial distribution models number of successes in an $n$ tosses of a coin with probability that toss $i$ ends with a success as $p$. The problem with a die can also be modeled like this, we will just treat $1$ (or $2$) as a success and remaining 5 numbers as a failure, which mathematically equivalent to saying we are tossing a coin with probability of success = $1/6$.
Regarding joint PMF, $\mathbb{P}[X = x, Y = y]$, you have ${30 \choose x}$ ways to select positions for $x$ 1s and ${30 - x \choose y}$ ways to select positions for $y$ 2s. Of course, we assume $x + y \leq 30$. The distribution you are after is then $$\mathbb{P}[X = x, Y = y] = {30 \choose x}{30 - x \choose y}(\frac{1}{5})^{x}(\frac{1}{5})^y(\frac{4}{6})^{30 - x - y}$$ which is indeed trinomial distribution.