Question:
How many chords can be drawn through 21 points on a circle?
Solution:
Number of points=21
⇒n=21
A chord connects circle at two points.
⇒r=2
$\Rightarrow$ Number of chords from 21 points $={ }^{n} C_{r}$
$\Rightarrow{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}={ }^{21} \mathrm{C}_{2}$
$\Rightarrow{ }^{21} C_{2}=\frac{21 !}{(21-2) ! \times 2 !}$
$\Rightarrow{ }^{21} \mathrm{C}_{2}=\frac{21 !}{19 ! \times 2 !}$
$\Rightarrow{ }^{21} C_{2}=\frac{21 !}{19 ! \times 2 !}$
$\Rightarrow{ }^{21} C_{2}=\frac{21 \times 20 \times 19 !}{19 ! \times 2 !}$
$\Rightarrow{ }^{21} \mathrm{C}_{2}=\frac{21 \times 20}{2 \times 1}$
$\Rightarrow{ }^{21} \mathrm{C}_{2}=210$ chords.
Ans: 210 chords can be drawn through 21 points on a circle.