Number of disjoint regions = number of intersections + 1 If I have N circles, where every circle intersects every other circle at 2 distinct points, then the number of disjoint regions equals the number of intersections + 1.
I would like to know if there is a nice intutive reasoning to why this is true.
edit: To clear up, every circle intersects every other circle and every pair of intersection points is unique. So you cannot have something like this

 A: Hint. Think inductively. Suppose you have a configuration with $n$ circles and the right number of regions and intersections. Now add another circle. How many new intersections are there? How have new regions been created?
Start by drawing a fourth circle in your three circle picture.
Edit in response to comment.
Suppose the answer is right for $n$ circles. Imagine the next one. It meets each of the old circles twice, so there are $2n$ new intersections, all on the new circle. Those intersections divide the new circle into $2n$ arcs. Each of those arcs cuts an old region in two, so there are $2n$ new regions. That says the answer is right for $n+1$ circles.
You can get the result directly from Euler's formula:
Suppose there are $n$ circles. Each meets the other $n-1$ twice, so contains $2(n-1)$ vertices that cut it into that many arcs. The total count of arcs is thus $2n(n-1)$. The total count of vertices is half that since each vertex appears on two circles.  Then
$$
1 = V - E + F = n(n-1) - 2n(n-1) + F
$$
so
$$
F = 1 + n(n-1) = 1 + V.
$$
A: Well, Let $a_n$=Number of Regions for number of circle drawn till n step
Define $b_n$=Increment of number of regions from $a_{n-1}$ to $a_n$
Thus $b_{n+1}-b_{n}$ is actually constant because speed of increment does not depend on n as it means the new circle cuts each of the previous regions into 2, implying $b_n=2(n-1) $ thus But $b_n=a_{n}-a_{n-1}=2(n-1)$ thus Thus $a_n={n(n-1)}+1$
Now, for $n$  circles we get two distinct points for each pair of circle so total number of point is $2$$n\choose 2$$=\frac{n(n-1)}{2}$ $\blacksquare$
N.B: This is a very philosophical argument think it deeply it took me about 4hrs to visualize.
Thanks to 60q for his comment!!!!
