Question

In a polygon no 3 diagonals are concurrent. If the total number of points of intersection of diagonals in the interior of the polygon is 70, find the number of diagonals.

Answer 1

$$\begin{gathered} DearStudent, \\ Letnumberofsidesofthepolygonben \\ Asselectionoffourverticesofthepolygongivesaninteriorintersection \\ \Rightarrow {}^{n}C_4=70 \\ \Rightarrow n(n-1)(n-2)(n-3)=24\times 70 \\ \Rightarrow n(n-1)(n-2)(n-3)=8\times 7\times 6\times 5 \\ \Rightarrow n=8 \\ \Rightarrow Numberofdiagonals={}^{8}C_2-8=\frac{8\times 7\times 6!}{2\times 6!}-8=28-8=20 \\ Regards \end{gathered}$$