1993 AIME Problem 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 suitable rotation, a brick of dimensions a_1 \times a_2 \times a_3 can be enclosed in a box of dimension 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?

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.