Question

Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords form a convex quadrilateral?

$\dfrac{1}{15}$

$\dfrac{1}{91}$

$\dfrac{1}{273}$

$\dfrac{1}{455}$

$\dfrac{1}{1365}$

Solution:

There are $\binom{6}{2} = 15$ chords, so $\binom{15}{4} = 1365$ ways to select four of them. A convex quadrilateral arises exactly when the four chords are the sides of a quadrilateral on four of the six points, and each choice of $4$ points gives exactly one such quadrilateral.

Hence there are $\binom{6}{4} = 15$ favorable outcomes, and the probability is $\dfrac{15}{1365} = \dfrac{1}{91}.$

Thus, the correct answer is B.

1999 AMC 12 Problem 24

Solution:

Problem 24 in Other Years