JEE Advance - Mathematics (2014 - Paper 1 Offline - No. 1)
Let $${n \ge 2}$$ 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
Antwort
5
Erläuterung
Number of blue lines $=n=$ number of sides of polygon so formed.
Number of red lines $={ }^n C_2-n$.
Thus, by joining $n$ points (not more than 2 on a line) there are ${ }^n C_2$ lines formed because for each line two points are required.
Also, red lines come after excluding sides of polygon.
Therefore, $n={ }^n C_2-n$
or ${ }^n C_2=2 n$
$\frac{n(n-1)}{2}=2 n$
or $n-1=4 (\because n \neq 0)$
$\therefore n=5$