Let n \geq 2 be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of n is

Engineering

Mathematics

Circular Permutation

Question

Let n $\geq$ 2 be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of n is

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$\underset{red lines}{\underset{⏟}{^{n} C_{2} - \underset{blue lines}{\underset{⏟}{n}}}} = \underset{blue lines}{\underset{⏟}{n}}$

⇒ n = 5

Aliter: Let there are n points on the circle.

⇒ Number of blue line = n

For red line, choose any point on the circle by ⁿC₁ ways (says P₁),

now remove pn and p₂. So number of points left = n – 3

$∴$ Number of red line = $\frac{^{n} C_{1} \cdot^{n - 3} C_{1}}{2}$

$∴ n = \frac{^{n} C_{1} \cdot^{n - 3} C_{1}}{2}$ = 5.