We have venn diagram and is it possible to write data as combinations? i.e k from n objects we have for example this diagram. I think we can

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  • $\begingroup$ It is very unclear what it is you mean by this. I assume that the numbers indicate how many elements the various sets have in common. You can certainly list this information out manually. You can even come up with some notational short-hands in that list. But if you mean express it somehow with $n\choose k$, then no. That notation is the number of different ways of choosing $k$ objects from $n$ objects. That is not information the Venn diagram is conveying. That value is irrelevant to this diagram. $\endgroup$ – Paul Sinclair Feb 23 '17 at 2:24

A Venn diagram is useful for representing intersecting sets. We can define a Venn diagram where:

  • the elements are the subsets of a set $S$,
  • the sets in the Venn diagram are sets of subsets containing a given element, i.e., $\{T \subseteq S:s \in T\}$ for each $s \in S$.

When $S=\{1,2,3\}$, the Venn diagram looks like:

Venn diagram

And $\binom{3}{2}$ counts the number of regions which intersect exactly two sets.

This is probably the closest useful relationship between "combinations" and "Venn diagrams" you're going to get. Venn diagrams are defined according to intersections, whereas combinations count $k$-element subsets: $$ \big\{\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\}\big\} $$ and $$ \big\{\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\}\big\} $$ for example. These subsets have empty intersections, but "intersections" is the defining feature of the Venn diagram.

A more appropriate tool is the lattice, where combinations can be viewed as the number of subsets at a given "level".

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