Probability each table leg was in each spot.

Suppose the legs are numbered and ordered in a list, for example [1,2,3,4,5,6]. When packed away the first three legs in the list are on the left, the last three are on the right. When erected the first two are on the left, the middle two in the middle and the last two on the right.

Let's consider the possible permutations that occur between a table being packed away and then re-erected. If we start with a table with the order [1,2,3,4,5,6], the possible orders after being packed are [1,2,3,4,5,6] or [1,2,4,3,5,6]. After being reassembled the 18 possible permutations are:

P = { [1,2,3,4,5,6], [2,3,1,4,5,6], [2,3,1,4,5,6], [1,2,3,5,4,6], [1,3,2,5,4,6], [2,3,1,5,4,6], [1,2,3,6,4,5], [1,3,2,6,4,5], [2,3,1,6,4,5], [1,2,4,3,5,6], [1,4,2,3,5,6], [2,4,1,3,5,6], [1,2,4,5,3,6], [1,4,2,5,3,6], [2,4,1,5,3,6], [1,2,4,6,3,5], [1,4,2,6,3,5], [2,4,1,6,3,5]}

As some commenters mentioned, working out precisely the expectation that all legs end up in all spots is (probably) very hard, but I think this is a great example of a situation where running randomised simulations in a computer should give a pretty good real world answer.

For a given number of disassemblies and erections between 1 and 100, I ran 10000 simulations and counted for how many simulations each of the legs occurred in each of the positions. The results:

So, if you have setup the table 100 times I would say it's very likely indeed (> 99%) that all the legs would have been in all positions!