# Number of intersections in a venn diagram with 8 circles

I am trying to calculate the number of intersections one would have in a Venn diagram with 8 overlapping circles but do not know where I would start.

Any help with the number and how you got there would be appreciated.

Edit: It could be a Euler diagram too - basically I'm trying to present the argument to a journal editor that the upset plot that I used is far superior to a Venn/Euler diagram because of the sheer number of intersections one would have to look at.

• It depends on how the circles overlap. Do you want all possible intersections, or do you have some specific configuration in mind? And do you want the number of regions, or the number of intersection points of the circles edges? Commented Oct 16, 2021 at 18:22
• Hint: Whenever you include a new "circle", it must pass into and out of every existing region. Consider this for the case of 2 circles (have 4 regions) to 3 circles (have ?? many regions), then 3 circles to 4 circles, etc. (Note: You can't show all possible regions using circles when there are 4 or more categories, so ellipses, rectangles, etc. are needed.) Commented Oct 16, 2021 at 18:23
• There is no Venn diagram with 4 or more circes, in the sense that all possible $2^n$ sets determined by whether or not an element is within each of the $n$ sets or not have connected regions to go with them. Commented Oct 16, 2021 at 18:25
• Thank you for answers! It could be a Euler diagram too. I'm basically trying to make the case to a journal editor that an Upset plot is vastly superior to a Venn/Euler given the number of intersections I'm working with. I'll edit that in if that's ok. Commented Oct 16, 2021 at 18:28
• @coffeemath See my link above. Commented Oct 16, 2021 at 18:51

Calculating the number of total intersections among n Venn diagrams involves summing the number of ways 2,3, ..., n circles intersect each other. Ask yourself:

• How many ways do two circles overlap?
• How many ways do three circles overlap?
• ...
• How many ways do $$n$$ circles overlap?

The answer to each of these questions is:

- n choose 2 - n choose 3 - ... - n choose n

Therefore, we can express the number of intersections among n Venn diagrams as a formula with the following summation: $$I(n) = \sum_{i = 2}^{n} \frac{n!}{i!(n-i)!}$$ $$I(8) = 247$$ So a Venn diagram with 8 overlapping circles has 247 intersections.

• Fantastic answer! I knew there had to be a rule, but couldn't think of it. Thank you! Commented Oct 18, 2021 at 5:43

Number of overlappings of r circles out of n circles is given by combinations choosing r from n ,that is r choose n = binomial (n,r) = C(n,r) When you consider n number of circles or sets on venn diagram, Intersections of 2 sets that is numberof overlappings of 2 circles = C(n,2) Intersections of 3 sets = C(n,3) ......................... .......................... Intersections of n sets = C(n,n) Total number of intersections = C(n,2)+C(n,3)+.......C(n,n) = {C(n,0)+C(n,1)+C(n,2)+........C(n,n)} - 1 -n = 2^n - n -1 Here you can obtain C(n,0)+C(n,1)+...C(n,n) by substituting x =1 to the binomial expansion of (1+x)^n

Now in your case since n =8 Total number of intersections is 247.