The formula there seems ok, although the notation is ambiguous so it may be hard to determine its meaning.

Suppose there are opponents with stacks of size 2, 3, 5, 10, 20, and your stack is of size 15. According to the ICM, the chance you finish 4th, so that the first 3 players have stacks of size 20, 2, and 5, respectively, is

(20/T)*(2/(T-20))*(5/(T-22))*(15/(T-27))

where T is the total number of chips in play 2+3+5+10+20+15 = 55.

To determine your chance to finish 4th according to the ICM, you would sum over all of the possible permutations of the first 3 players. There would be 5*4*3 = 120 choices for first, second, and third. For each of these, you get a product like this, and you add up the 120 probabilities.

You do not need to do ICM calculations while playing. For most purposes when you are not playing, you can just use an ICM calculator which has already been written, such as the one built into SNG Wizard or my program

ICM Explorer. Actually, the one in ICM Explorer uses a different algorithm which is faster so that it can calculate the probability of finishing 10th without doing 10*9*8*7...*2 = 3628800 calculations. The algorithm in ICM Explorer only needs a few hundred calculations instead.