Question

How many triangles can be formed by joining four points on a circle?
A. 4
B. 6
C. 8
D. 10

Answer 1

Hint: We first establish the fact that the no three points out of those four points in the circle is collinear. We use the combinatorial formula of ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$ to find the number of ways we can choose 3 points out of 4 points.

Complete step-by-step solution:
We have to find the number of triangles that can be formed by joining four points on a circle.
As the points are on a circle, it is clear that no three points out of those four points are collinear.
We are going to use the concept of permutation to find the number of triangles.
To form a triangle, we need in total three non-collinear points as they create three sides of a triangle.
There are four points and we have to choose three out of them.
We know that the number of ways we can choose r things out of n things where they are not alike is ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$.
Here we have to choose 3 points out of 4 points which can be done in ${}^{4}{{C}_{3}}=\dfrac{4!}{3!\left( 4-3 \right)!}=4$ ways.
The correct option is A.

Note: The number of vertices, sides and angles are the same for a triangle. Therefore, the number of vertices decides the number of sides which help in combinational evaluation. If we had to find the diagonals in that case, we had to subtract the number of sides.