Number of restricted arrangements

I am looking for a faster way to solve this problem:

Let's suppose we have n boxes and n marbles (each of them has a different kind). Every box can contain only some kinds of marbles (It is shown it the example below), and only one marble fits inside one box. Please read the edits.

The question is: In how many ways can I put marbles inside the boxes in polynomial time?

Example:

n=3

Marbles: 2,5,3

Restrictions of the i-th box (i-th box can only contain those marbles): {5,2},{3,5,2},{3,2}

The answer: 3, because the possible positions of the marbles are: {5,2,3},{5,3,2},{2,5,3}

I have a solution which works in O(2^n), but it is too slow. There are also one limitation about the boxes tolerance, which I don't find very important, but I will write them also. Each box has it's own kind-restrictions, but there is one list of kinds which is accepted by all of them (in the example above this widely accepted kind is 2).

Edit: I have just found this question but I am not sure if it works in my case, and the dynamic solution is not well described. Could somebody clarify this? This question was answered 4 years ago, so I won't ask it there. https://math.stackexchange.com/questions/2145985/how-to-compute-number-of-combinations-with-placement-restrictions?rq=1

Edit#2: I also have to mention that excluding widely-accepted list the maximum size of the acceptance list of a box has 0 1 or 2 elements.

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