JEE Advanced · 2014 · 25 May 2014 · Paper 1 · Permutations and Combinations

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

Answer: 5

Solution

The total number of line segments joining

n

points is

(2 n) = \frac{n ( n - 1 )}{2}

. The number of blue segments (joining adjacent points on the circle) is

n

. The number of red segments is

\frac{n ( n - 1 )}{2} - n = \frac{n ( n - 3 )}{2}

. Setting red equal to blue:

\frac{n ( n - 3 )}{2} = n

, which gives

n (n - 3) = 2 n

, so

n - 3 = 2

, hence

n = 5

JEE Advanced · 2014 · 25 May 2014 · Paper 1 · Integer answer

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