Example 5. $n$ circles on a plane, any two of which intersect at two points, and no three circles pass through the same point, divide the plane into how many regions?

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Solve for the number of intersection points $p=2 C_{n}^{2}$, thus the number of regions the plane is divided into is $S=2 C_{n}^{2}+2=n^{2}-n+2$.