Question

There are n points on a circle. The number of straight lines formed by joining them is equal to
a.${}^{n}{{C}_{2}}$
b.${}^{n}{{P}_{2}}$
c.${}^{n}{{C}_{2}}-1$
d.None of these

Answer 1

Hint: Mathematics includes the study of topics which are related to quantity, structure, space and change. For our problem, we are provided with n points on a circle. To figure out the number of straight lines formed we must employ the combination property of lines passing through points on a circle.

Complete step-by-step answer:
Mathematics related to all the phenomena occurring in the world. When mathematical structures are good models of real phenomena mathematical reasoning can be used to provide insight or predictions about nature. There are several types of operators which are available in mathematics. These operators perform the basic function of doing calculation with certain numeric data. Most basic types of operators are addition, subtraction, multiplication and division.

In addition, we add two numbers. By using these operators, we can formulate formulas for combination which can be stated as: “The number of ways of selecting r items or objects from a group of n distinct items or objects is
${}^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}$

Now, we are given a circle with n points on it.
This implies that number of distinct items = n.

A straight line is formed by joining two points. This implies that we require two points at a time.

Therefore, the number of ways of selecting items, r = 2.

This implies that putting these values in combination formula we get,
$\dfrac{n!}{2!(n-2)!}={}^{n}{{C}_{2}}$

This implies that the correct answer is option (a).

Note: The key step for solving this problem is the knowledge of combination of selecting r items from a group of n distinct objects. In this way we can obtain the possible number of straight lines that passes through two given points in a circle. By using this methodology our problem is easily solved.