Interesting results SmoiL. Yeah your results may well be correct and mine wrong. Like I said I only did an extremely quick and nasty job of coding it. Oh well at least this makes it interesting, two different result and neither sure exactly who is correct yet.
Just to make it clear of what I was calculating, my program was initially just written to take one four digit number as input and then to calculate whether or not that number had the "24'ness" property and write this as the output. After testing it on a few numbers and seeing that it seemed to work I thought it would be interesting the run all numbers from 0 to 9999 through in a loop and count how many satisfied the condition. I was interested in the probability that a randomly chosen four digit number would satisfied the condition.
What SAO is interested in is a different thing (I know you understand this). He is interested in the number of unique sets of four decimal digits that satisfy the condition.
BTW, the total number of such sets (not just the ones that satisfy the condition but the total number of unique sets) is C(10,4) + 3*C(10,3) + 3*C(10,2) + C(10,1) = 715, where C(n,r) is the combination function, n!/(r! (n-r)!).
I know how to enumerate these sets pretty easily but dont have time today. If I get time I'll get my program to enumerate just these set tomorrow and set what the satisfy count is.
PS. My crude program was a console application that just spat out results to a command prompt window. I was only looking at the final count but next time I run it I'll make it log the results to a file so we can scrutinize the individual numbers as per your list file.