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:
Two points are joined to draw a single line. Hence, the number of lines can be formed is the combination of selecting 2 points from $n$ points on the circle.

Complete step by step solution:
We have been given that there are $n$ points on a circle.
We know that we need two points to draw a line.
Therefore, we have to select 2 points from $n$ points on a circle to draw a line.
Since the order of the points does not matter, we will use the concept of combination to select 2 points from a given number of points.
The number of ways in which $r$ objects can be selected from $n$ objects is given by ${}^n{C_r}$
On substituting the value of $r$ as 2, we get the number of lines that can be drawn using the $n$ points is \[{}^n{C_2}\].
Thus, the number of lines formed by joining $n$ points is equal to \[{}^n{C_2}\].

Hence, option A is correct.

Note:
Every line formed by joining 2 points will be unique. Also, the order of the points does not matter, and hence the combination is used. If there is a line $AB$ whose end-points are $A$ and $B$. Then, the line $AB$ and $BA$ represents the same line. And the value of $^n{C_r}$ is equal to $\dfrac{{n!}}{{r!\left( {n - r} \right)!}}$.