A question in Permutations and combinations

[Vishalrox]

there are 4 circles and 4 straight lines...find the maximum number of intersecting points possible in the intersection of all these given figures...i can't get how to solve it...

 

[Vorde]

Do you mean the maximum number of points where they all intersect? Or just at least 2 intersect?

 

[mesa]

A circle can intersect another circle only twice so you would have:
6 intersects + 4 intersects + 2 intersects= 12

A straight line can intersect a circle a maximum of twice and there are four circles so:
8 intersects per line x 4 lines = 32 intersects.

Each line can also intersect the other lines at a single point so they overlap each of the four lines so not recounting an intersect would be:
3+2+1=6

Add them up:
12+32+6= 50 max intersects

Someone should check this ;)

 

[haruspex]

A circle can intersect another circle only twice so you would have:
6 intersects + 4 intersects + 2 intersects= 12

A straight line can intersect a circle a maximum of twice and there are four circles so:
8 intersects per line x 4 lines = 32 intersects.

Each line can also intersect the other lines at a single point so they overlap each of the four lines so not recounting an intersect would be:
3+2+1=6

Add them up:
12+32+6= 50 max intersects

That's certainly an upper bound, but it's not immediately obvious that all these intersections are achievable simultaneously.
Start with some circle C. (This is not one of THE circles, it's just used for construction.)
You can arrange N equal circles, larger than C, such that each surrounds C. Clearly each pair of these intersects.
Any M straight lines in general position must intersect each other.
The region in which the intersections of the lines occurs can be bounded by a circle, D. Shrink that picture as necessary and fit D inside C. Necessarily, every line intersects every circle.
So any number of circles and lines can be arranged to achieve the upper bound.