Question 7
Three numbers, $a_1,$ $a_2,$ $a_3,$ are drawn randomly and without replacement from the set $\{1,2,3,\ldots,1000\}.$ Three other numbers, $b_1,$ $b_2,$ $b_3,$ are then drawn randomly and without replacement from the remaining set of $997$ numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1\times a_2\times a_3$ can be enclosed in a box of dimensions $b_1\times b_2\times b_3,$ with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?