is it possible to have a venn diagram represent de Montmort matching problem where the number of cards is $n=3$ with the elements included in the diagram?

I understand how inclusion/exclusion works(thanks to the kind contributors here) with $ \bigcup_{i=1}^n A_i = \sum_{i=1}^n A_i - \sum_{i<j} A_i \cap A_j + \sum_{i<j<k} A_i \cap A_j \cap A_k - \dots + (-1)^{n+1} A_i \cap \dots A_n$.

However, i can't picture where the elements are placed in the venn diagram with 6 permutations of $n=3$. Feel free to expand the $n$ value if it helps represent the matching problem via the venn diagram.

Kindly advise. Thank you

added* description of Montmort’s matching problem

"Let there be $n$ objects numbered from 1 to $n$, and let them be ordered at random, assuming that the $n!$ permutations are equally probable. A coincidence occurs if object number $i$ is found at the $i$th place. The problem is to find the number of permutations with at least one coincidence or, equivalently, the probability of at least one coincidence...:"[A. Hald]

quoted from rstudio

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    $\begingroup$ You need to tell us what Montmort's matching problem is (by editing, not in the comments), because this is not common knowledge. $\endgroup$ Apr 15, 2022 at 15:44
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    $\begingroup$ Seems you are asking about a derangement problem. And Venn Diagram is used to illustrate the relationship between various set - in probability we call them events. So you need to first determine the events you interested in to be shown in this Venn diagram. E.g. The events of card $1,2,3$ being matched. And then you place the singletons inside. $\endgroup$
    – BGM
    Apr 15, 2022 at 20:28
  • $\begingroup$ @ BGM, thank you for the reply. This is where is gets cloudy for me, there are $n=3$ with $n!$ permutations. If we use "pebble world" to describe the sample space, there are 3 pebbles and 6 sets within the space. [('1', '2', '3'), ('1', '3', '2'), ('2', '1', '3'), ('2', '3', '1'), ('3', '1', '2'), ('3', '2', '1')]. All the sets are overlapping, making it confusing for me to picture it in a venn diagram, hence, the question. $\endgroup$ Apr 16, 2022 at 5:40
  • $\begingroup$ @Mike Earnest, thank you for the reply, i've added a shot description as suggested. $\endgroup$ Apr 16, 2022 at 5:53

1 Answer 1


Here is the finest Venn diagram that Windows paint can offer:

enter image description here


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