Question

Find the maximum number of points of intersection of five lines and four circles .

Answer 1

The point of intersection of two circles is at two distinct points.
The point of intersection of two straight lines is at one point.
The point of intersection of $1$ circle and one straight line intersect at two distinct points.
The total numbers of points of intersections are
$\begin{array}{l}Twostraightlines{:}^5{\mathrm{C}}_2{\Rightarrow }^5{\mathrm{C}}_2\times 1=10\\ Twocircles{:}^4{\mathrm{C}}_2{\Rightarrow }^4{\mathrm{C}}_2\times 2=12\\ Onelineandonecircle{:}^5{\mathrm{C}}_1{\times }^4{\mathrm{C}}_1{\Rightarrow }^5{\mathrm{C}}_1{\times }^4{\mathrm{C}}_1\times 2=40\\ Total=62\end{array}$

Hence, the maximum number of points of intersection are $62$