Question

Find the number of triangles obtained by joining 10 points on a plane if
1) No three of which are collinear.
2) Four points are collinear.

Answer 1

Hint:
A triangle is formed by joining 3 points, this is a concept of combination. A triangle can be formed when only 2 points are collinear, if more than 2 points are collinear then the triangle cannot be formed.

Complete step by step solution:
Before going into the solution, let us be clear about the concept
Permutation: Arranging the numbers in order is called permutation, the formula of permutation is \[^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\]
Where n= Total number of items in the sample, r= number of items to be selected from the sample.
Combination: Selecting the items from the sample is called combination, the formula of combination is \[^n{C_r} = {\dfrac{{n!}}{{r!\left( {n - r} \right)!}}_r}\]
Where n= Total number of items in the sample, r= number of items to be selected from the sample.
1) No three of which are collinear
Given that no three of which are collinear, which means no three points are not in a straight line which means out of 10 points, the number of triangles can be formed in $^{10}{C_3}$ ways.
Number of triangles that can be formed when no three of which are collinear are,
\[
  { \Rightarrow ^{10}}{C_3} \\
   \Rightarrow \dfrac{{10!}}{{3!\left( {10 - 3} \right)!}} \\
   \Rightarrow \dfrac{{10!}}{{3!7!}} \\
   \Rightarrow \dfrac{{10 \times 9 \times 8 \times 7!}}{{3!7!}} \\
   \Rightarrow \dfrac{{10 \times 9 \times 8}}{{3 \times 2 \times 1}} \\
   \Rightarrow 120 \\
 \]
The number of triangles that can be formed is in 120 ways.
2) Four points are collinear.
Given that four points are collinear, which means four points are in the straight line, whereas a triangle needs three points to form a triangle.Out of 10 points, 4 points cannot form a triangle. So, the number of triangles formed when four points are collinear are in $^{10}{C_3}{ - ^4}{C_3}$ways.
Number of triangles that can be formed when four points are collinear are
\[
  { \Rightarrow ^{10}}{C_3}{ - ^4}{C_3} \\
   \Rightarrow \dfrac{{10!}}{{3!\left( {10 - 3} \right)!}} - \dfrac{{4!}}{{3!\left( {4 - 3} \right)!}} \\
   \Rightarrow \dfrac{{10 \times 9 \times 8 \times 7!}}{{3!7!}} - \dfrac{{4 \times 3 \times 2 \times 1!}}{{3!1!}} \\
   \Rightarrow \dfrac{{10 \times 9 \times 8}}{{3 \times 2 \times 1}} - \dfrac{{4 \times 3 \times 2}}{{3 \times 2 \times 1}} \\
   \Rightarrow 120 - 4 \\
   \Rightarrow 116 \\
 \]$$

The number of triangles that can be formed when Four points are collinear are 116.

Note:
When points are collinear, a triangle cannot be formed, unless if only two points are collinear. When all points are in collinear no triangle can be formed. Because it won’t be having any area. That’s why we use it to show the area of a triangle is zero to prove points are collinear.