How to solve this task about circles and lines intercepting each other?

Question

We have drawn some lines and circles on a paper. Every two has an interception, but none three goes through the same point. How many lines and circles have we drawn if we have 75 interceptions?

Answer 1

Suppose you have $m$ lines and $n$ circles. Any pair of lines has exactly one intersection. I assume that any pair of circles as well as any combination of one line and one circle has two intersections, since otherwise this question would be impossible to answer. So you have in total

$$\binom{m}{2} + 2mn + 2\binom{n}{2}$$

points of intersection. Since $\binom{13}{2}=78>75$ you can be sure that $m,n<13$. Simple trial and error will show you that in the range $0\le m,n<13$ there is only one pair satisfying the number of intersections given in your problem statement. If you don't want to (let a computer) compute all counts for all pairs in that range, you can start with $n=12,m=0$. Whenever the count is too low, increase $m$. Whenever it is too high, decrease $n$. Do so until you have found your solution.