Morrison on Metrics: The mathematics of permutations

Software choices are just one example of how combinatorial math can affect legal teams

The more ideas and possible mixes and matches a law department manager has available to choose from, the more permutations there are. Obvious, yes, but what are the mathematics?

Let's consider an example from software. According to the General Counsel Metrics Insights report on matter management software, there are at least 15 different packages that five or more U.S. law departments have licensed. Let’s assume any of them would be a potential choice for a given department. On the contract management software front, there are at least 15 software packages that the same law department can license. (There are more than this of both kinds of software, but the point being made still holds.).

If furthermore there are 10 choices for software applications that manage corporate entities such as subsidiaries and their boards and corporate minutes, and another 10 choices for patent tracking applications, think of all of the permutations. That general counsel could pick and choose one each from among those 50 packages, and the variations that could result are myriad.

The math to figure out the range of mixes and matches is straightforward. With those four genres of software, and a general counsel who is free to choose any of them independently of any other, there are 15x15x10x10 possible arrangements—22,500 combinations.

That formidable number quickly grows larger if you add in even more software choices that are not uncommon in legal departments, such as document management (if there are 10 choices the range of possibilities jumps 10 times larger). No wonder IT departments that support the legal team deeply appreciate short lists of candidates. And no wonder the software profile of departments turn out so differently.

Combinatorial math, however, goes beyond multiplicative increases. Think about all the steps and tasks that could be done to select and implement software. Let’s assume there are 10 tasks. For simplicity, let’s assume a department could do those tasks in any order. The number of different ways to arrange those tasks can be computed easily: 10 times 9 times 8 times 7 … The result is a whopping 3,628,800 different arrangements of tasks. This is called 10 factorial, which mathematicians write as “10!”.

Or, as another example, assume that a general counsel can choose among the six kinds of law department software mentioned above. The catch is she has to decide in what order to install them. If each could be implemented independently of the others, that creates 6 times 5 times 4 times 3 times 2 configurations. Excel tells you with the function “=fact(6)” (6 factorial) that the possibilities are ample: 720.

Some people refer to combinations as permutations—all the ways a set of independent things can be rearranged. Multiplying the choices works if you are simply shifting their order, but combinatorial math with factors comes into play if each choice reduces the subsequent choices. Combinatorial functions actually grow faster than exponential functions above some relatively modest number.

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