Joint distributions of the rvs $X_1,X_2,X_3$ which are exchangeable and only take values in $\{0,1\}$ Problem: Describe all possible joint distributions of the random variables $X_1,X_2,X_3$ which are exchangeable and only take values from $\{0,1\}.$ 

Thoughts: First let me note that I have previously posted a similar question: Joint distributions of the rvs $X_1,X_2$ which are exchangeable and only take values from $\{0,1\}$. Therefore, I am trying to apply the method of Parcly in this problem. Hence, let $P(X_1=0,X_2=0,X_3=0)=a$ and $P(X_1=1,X_2=1,X_3=1)=b$. I have the following theorem:
If $X_1,\dots,X_n$ are discrete exchangeable random variables, then $$P(X_1=x_1,\dots,X_n=x_n)=P(X_1=x_{\sigma(1)}\dots,X_n=x_{\sigma(n)})$$
for all permutations $\sigma$ on $\{1,\dots,n\}$ and for all choices of all real numbers $x_1,\dots,x_n.$ However, in the current problem we only have available choices of the real numbers $x_1,x_2,x_3$ form the set $\{0,1\}$, which would lead to a nonbijective permutation.
Does anybody have any hints on how to get around this issue?
Thank you for your time and highly appreciate any feedback.

This time you have $3$ degrees of freedom. The probabilities are described by $4$ distinct variables, $p_n=P(n\text{ of the }X_i\text{ are true})$ for $n=0,1,2,3$, and that all probabilities sum to $1$ is encapsulated in the constraint $p_0+3p_1+3p_2+p_3=1$. So this easily leads to a description of all distributions.
The generalisation to $n$ exchangeable random variables follows.
