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

Question

Vedantu Content Team · Accepted Answer

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 getNumber 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.