## Problem 1

If six points are chosen sequentially at random on the circumference of a circle, what is the probability that the triangle formed by the first three is disjoint from that formed by the second three.


## Solution

Answer: $3 / 10$.

Only the order is important. We are interested in permutations of 123456 where the 123 are together (allowing wrapping). wlog the 1 is in first position. So the triangles are disjoint in the cases $123 \mathrm{xxx}, 132 \mathrm{xxx}$, $1 \mathrm{xxx} 23,1 \mathrm{xxx} 32,12 \mathrm{xxx} 3,13 \mathrm{xxx} 2$. So the probability is $6 \cdot 6 / 5!=3 / 10$.

