Question

How many chords can be drawn through 21 points on a circle?

Answer 1

Hint:
Here, we will use the concept of chord of a circle. A chord is defined as a line segment formed by joining any two distinct points lying on the circle. It is formed by joining any 2 of the 21 points lying on the circle. We will use the formula for combinations to find the required number of chords.
Formula Used: The number of ways in which the \[n\] objects can be placed in \[r\] spaces, where order of objects is not important, can be found using the formula for combinations \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\].

Complete step by step solution:
We will use combinations to find the number of chords that we can draw through 21 points on a circle.
The number of ways in which the \[n\] objects can be placed in \[r\] spaces, where order of objects is not important, can be found using the formula for combinations \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\].
The number of chords that can be formed using the 21 points, taking 2 points at a time, is given by \[{}^{21}{C_2}\].
Substituting \[n = 21\] and \[r = 2\] in the formula for combinations, we get
Number of chords that can be formed using the 21 points \[ = {}^{21}{C_2} = \dfrac{{21!}}{{2!\left( {21 - 2} \right)!}}\]
We know that \[n!\] can be written as \[n\left( {n - 1} \right)!\] or \[n\left( {n - 1} \right)\left( {n - 2} \right)!\].
Rewriting \[21!\] as \[21\left( {21 - 1} \right)\left( {21 - 2} \right)!\], we get
\[ \Rightarrow \]Number of chords that can be formed using the 21 points \[ = {}^{21}{C_2} = \dfrac{{21\left( {21 - 1} \right)\left( {21 - 2} \right)!}}{{2!\left( {21 - 2} \right)!}}\]
Simplifying the expression, we get
\[ \Rightarrow \]Number of chords that can be formed using the 21 points \[ = \dfrac{{21 \times 20 \times 19!}}{{2 \times 1 \times 19!}}\]
Therefore, we get
\[ \Rightarrow \]Number of chords that can be formed using the 21 points \[ = 21 \times 10\]
Multiplying 21 by 10, we get
\[ \Rightarrow \] Number of chords that can be formed using the 21 points \[ = 210\]

Therefore, the number of chords that can be formed using the 21 points lying on the circle is 210 chords.

Note:
We used combinations instead of permutations to find the number of chords that can be formed using the points on the circle. A chord is formed using 2 points. Suppose two of the 21 points are A and B. Now, if A is joined to B, the chord is AB. If B is joined to A, the chord is BA. However, BA and AB are the same. If we used permutations, the chords AB and BA will be counted as 2 different chords. Therefore, we have used combinations since it does not matter which point is the initial points and which point is the terminal point.