Question

In a convex hexagon, two diagonals are drawn at random. The probability that the diagonals intersect at an interior point of the hexagon, is

• A. $\frac{5}{12}$
• B. $\frac{7}{12}$
• C. $\frac{2}{5}$
• D. None of these

Answer 1

The correct option is A $\frac{5}{12}$

In a convex hexagon there are 6 points.

So total number of ways to join any two points $={}^6{C}_2$

Out of which there are 6 edges and rest are diagonals.

$\therefore$ no of diagonals $={}^6{C}_2-6$

$=\frac{6!}{2!.4!}-6$

$=15-6$

$=9.$

Out of $9$ diagonals 2 diagonals can be chosen in ${}^9{C}_2=36$ ways.

Intersection will occur if two diagonal intersect i.e, if 4 points can joins.

$\therefore$ Total number of intersections $={}^6{C}_4=15$

$\therefore$ The probability that diagonal intersect at an interior point $=\frac{15}{36}=\frac{5}{12}$

Hence, the answer is $\frac{5}{12}.$