How many chords can be drawn in a circle?

Number of chords that can be formed using the 21 points =21×20×19! 2×1×19! Therefore, the number of chords that can be formed using the 21 points lying on the circle is 210 chords.

vedantu.com - How many chords can be drawn through 21 points on a circle?

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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} ight)!}}].

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} ight)!}}]. 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} ight)!}}] We know that [n!] can be written as [nleft( {n - 1} ight)!] or [nleft( {n - 1} ight)left( {n - 2} ight)!]. Rewriting [21!] as [21left( {21 - 1} ight)left( {21 - 2} ight)!], we get [ Rightarrow ]Number of chords that can be formed using the 21 points [ = {}^{21}{C_2} = dfrac{{21left( {21 - 1} ight)left( {21 - 2} ight)!}}{{2!left( {21 - 2} ight)!}}]

Simplifying the expression, we get

[ Rightarrow ]Number of chords that can be formed using the 21 points [ = dfrac{{21 imes 20 imes 19!}}{{2 imes 1 imes 19!}}]

Therefore, we get

[ Rightarrow ]Number of chords that can be formed using the 21 points [ = 21 imes 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.

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