# Counting in Geometry

(a) In a plane, if there are n points of which no three are collinear, then

- The number of straight lines that can be formed by joining them is
^{n}C_{2}. - The number of triangles that can be formed by joining them is
^{n}C_{3}. - The number of polygons with k sides that can be formed by joining them is
^{n}C_{k}.

(b) In a plane, if there are n points out of which m points are collinear, then

- The number of straight lines that can be formed by joining them is
^{n}C_{2}–^{m}C_{2}+ 1. - The number of triangles that can be formed by joining them is
^{n}C_{3}–^{m}C_{3}. - The number of polygons with k sides that can be formed by joining them is
^{n}C_{k}–^{m}C_{k}.

(c) The number of diagonals of an n sided polygon is ^{n}C_{2} – n = n × (n – 3)/2