From the set {1, 2, 3, ... 999}, 6 distinct numbers are chosen. These are divided into two groups $a_1,a_2,a_3$ and $b_1,b_2,b_3$. Find the probability that a brick made from the dimensions of group $a$ fits into a box made from the dimensions of $b$. Assume that the brick can be rotated in a suitable manner to be made to fit inside the box
I am able to comprehend this question and what it asks. However, I don't have a strategy in place to solve this question. The solution provided has the following to say:
without loss of generality, we can say that out of the 20 possible cases, 5 are suitable. Therefore, the probability is $\frac 1 4$.
What are the twenty possible cases here? How can we set conditions on $a$ and $b$ such that we can obtain these cases? The ability to rotate the brick before placing in the box has confused me. Any help would be appreciated!