Question

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

Answer 1

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.