Track / Stage 4 / 112 of 340 #372 of 2000

Problem 372

AMC 12 late, AIME early

Combinatorics Difficulty 4.8 Find the answer

15. (1999 American High School Mathematics Examination) Given six points on a circle, any combination of four chords can form a convex quadrilateral. The probability that they form a convex quadrilateral is ( ).
A. $\frac{1}{15}$
B. $\frac{1}{91}$
C. $\frac{1}{273}$
D. $\frac{1}{455}$
E. $\frac{1}{1365}$

Your answer

Official solution

15. B. Reason: The total number of chords that can be connected by the six given points is $C_{6}^{2}=15$ ; the number of ways to choose four chords from these is $C_{15}^{4}$ . According to the problem, the number of convex quadrilaterals that can be formed by the chords connecting the six points is $C_{6}^{4}$ , thus the required probability is
$P=\frac{C_{6}}{C_{15}}=\frac{1}{91} .$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.