It takes $3\times 2 \times 2 \times 3 = 36$ numbers to write down a probability distribution on all possible values of these variables. They are redundant, because they must sum to $1$. Therefore the number of (functionally independent) parameters is $35$.
If you need more convincing (that was a rather hand-waving argument), read on.
By definition, a sequence of such random variables is a measurable function
$$\mathbf{X}=(X_1,X_2,X_3,X_4):\Omega\to\mathbb{R}^4$$
defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. By limiting the range of $X_1$ to a set of three elements ("states"), etc., you guarantee the range of $\mathbf{X}$ itself is limited to $3\times 2\times 2 \times 3=36$ possible values. Any probability distribution for $\mathbf{X}$ can be written as a set of $36$ probabilities, one for each one of those values. The axioms of probability impose $36+1$ constraints on those probabilities: they must be nonnegative ($36$ inequality constraints) and sum to unity (one equality constraint).
Conversely, any set of $36$ numbers satisfying all $37$ constraints gives a possible probability measure on $\Omega$. It should be obvious how this works, but to be explicit, let's introduce some notation:
Let the possible values of $X_i$ be $a_i^{(1)}, a_i^{(2)}, \ldots, a_i^{(k_i)}$ where $X_i$ has $k_i$ possible values.
Let the nonnegative numbers, summing to $1$, associated with $\mathbf{a}=(a_1^{(i_1)}, a_2^{(i_2)}, a_3^{(i_3)}, a_4^{(i_4)})$ be written $p_{i_1i_2i_3i_4}$.
For any vector of possible values $\mathbf{a}$ for $\mathbf{X}$, we know (because random variables are measureable) that $$\mathbf{X}^{-1}(\mathbf{a}) = \{\omega\in\Omega\mid \mathbf{X}(\omega)=\mathbf{a}\}$$ is a measurable set (in $\mathcal{F}$). Define $$\mathbb{P}\left(\mathbf{X}^{-1}(\mathbf{a})\right) = p_{i_1i_2i_3i_4}.$$
It is trivial to check that $\mathbb{P}$ is an $\mathcal{F}$-measurable probability measure on $\Omega$.
The set of all such $p_{i_1i_2i_3i_4}$, with $36$ subscripts, nonnegative values, and summing to unity, form the unit simplex in $\mathbb{R}^{36}$.
We have thereby a established a natural one-to-one correspondence between the points of this simplex and the set of all possible probability distributions of all such $\mathbf{X}$ (regardless of what $\Omega$ or $\mathcal{F}$ might happen to be). The unit simplex in this case is a $36-1=35$-dimensional submanifold-with-corners: any continuous (or differentiable, or algebraic) coordinate system for this set requires $35$ numbers.
This construction is closely related to a basic tool used by Efron, Tibshirani, and others for studying the Bootstrap as well as to the influence function used to study M-estimators. It is called the "sampling representation."
To see the connection, suppose you have a batch of $36$ data points $y_1, y_2, \ldots, y_{36}$. A bootstrap sample consists of $36$ independent realizations from the random variable $\mathbf X$ that has a $p_1=1/36$ chance of equaling $y_1$, a $p_2=1/36$ chance of equaling $y_2$, and so on: it is the empirical distribution.
To understand the properties of the Bootstrap and other resampling statistics, Efron et al consider modifying this to some other distribution where the $p_i$ are no longer necessarily equal to one another. For instance, by changing $p_k$ to $1/36 + \epsilon$ and changing all the other $p_j$ ($j\ne k$) by $-\epsilon/35$ you obtain (for sufficiently small $\epsilon$) a distribution that represents overweighting the data value $X_k$ (when $\epsilon$ is positive) or underweighting it (when $\epsilon$ is negative) or even deleting it altogether (when $\epsilon=-1/36$), which leads to the "Jackknife".
As such, this representation of all the weighted resampling possibilities by means of a vector $\mathbf{p} = (p_1,p_2, \ldots, p_{36})$ allows us to visualize and reason about different resampling schemes as points on the unit simplex. The influence function of the value $X_k$ for any (differentiable) functional statistic $t$, for instance, is simply proportional to the partial derivative of $t(X)$ with respect to $p_k$.
Reference
Efron and Tibshirani (1993), An Introduction to The Bootstrap (Chapters 20 and 21).