Permutations and Combinations 2 Question 15
15. 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
(2014 Adv)
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Answer:
Correct Answer: 15. (5)
Solution:
- $\quad$ PLAN Number of line segment joining pair of adjacent point $=n$ Number of line segment obtained joining $n$ points on a circle $={ }^{n} C _2$
$$ \text { Number of red line segments }={ }^{n} C _2-n $$
Number of blue line segments $=n$
$$ \begin{aligned} \therefore & { }^{n} C _2-n & =n \\ \Rightarrow & \frac{n(n-1)}{2} & =2 n \\ \Rightarrow & n & =5 \end{aligned} $$
