Probability that values from different normal distributions will be in a given order



I am looking at several normal distributions that describe the same metric for different populations (independent). I want to find the probability that they are in a certain order.

For example, I have distributions A, B, C, and D. I want to know the probability that they are ranked from highest to lowest B,C,A,D (or any other permutation).

I have been trying to research it myself; however, I am not getting to a good answer. I think that this page would have something to do with the solution, but I don't know how to extend it to multiple distributions in a certain order: Probability of a point taken from a certain normal distribution will be greater than a point taken from another?

I have been attempting this for a while and keep getting incorrect results. As a sanity check I would find the probability of all permutations and sum them up. I was looking for them to equal 100%, but the closest I ever got was ~76%.

Any help would be greatly appreciated.