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

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.

 

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