Question

Find the number of chords that can be drawn through $16$ points on a circle.
\[1)\]$102$
\[2)\]$120$
\[3)\]$12$
\[4)\]${}^{16}{P_2}$

Answer 1

Hint: We have to find the number of chords that can be drawn through the given $16$ points on a circle . We solve this by using the concept of permutation and combination and the concept of chords on a circle . There are $16$ different points and $2$ points are required for a chord . By using the formula of the number of combinations of $n$ different things are taken $r$ at a time , denoted by \[{}^n{C_r}\].

Complete step-by-step answer:
A chord of a circle is defined as the line segment which joins two points such that the two points lie on the circle itself . The line joining the two points is known as the chord of the circle . A circle can have an infinite number of chords as a circle is a collection of an infinite number of points . The diameter of the circle is the longest possible chord of that particular circle .
Given :
Total number of points $ = 16$
Required points for a chord $ = 2$
The total number of chords which can be drawn $ = {}^{16}{C_2}$
Now , using the formula of \[{}^n{C_r}\]
\[{}^n{C_r}\] \[ = {\text{ }}\dfrac{{\left( {n!} \right)}}{{\left( {r!} \right) \times \left( {n - r} \right)!}}\]
Substituting the values in the formula , we get
${}^{16}{C_2}$ \[ = {\text{ }}\dfrac{{\left( {16{\text{ }}!} \right)}}{{2! \times \left( {16 - 2} \right)!}}\]
${}^{16}{C_2}$\[ = {\text{ }}\dfrac{{\left( {16{\text{ }}!} \right)}}{{2! \times 14!}}\]
${}^{16}{C_2}$$ = 120$
Hence , the number of chords that can be drawn on the circle $ = 120$
Thus the correct option is $(2)$
So, the correct answer is “Option 2”.

Note: Corresponding to each combination of \[{}^n{C_r}\] we have \[r!\] permutations, because $r$ objects in every combinations can be rearranged in \[r!\] ways. Hence , the total number of permutations of $n$ different things taken $r$ at a time is \[{}^n{C_r}\]\[ \times {\text{ }}r!\]. Thus \[{}^n{P_r} = \]\[{}^n{C_r}\]\[ \times {\text{ }}r!{\text{ }},{\text{ }}0 < {\text{ }}r{\text{ }} \leqslant n\;\]
Also , some formulas used :
\[{}^n{C_1}{\text{ }} = {\text{ }}n\]
\[{}^n{C_2}{\text{ }} = {\text{ }}\dfrac{{n\left( {n - 1} \right)}}{2}\]
\[{}^n{C_0}{\text{ }} = {\text{ }}1\]
\[{}^n{C_n} = {\text{ }}1\]