I just read a really interesting application of combinatorics to criminology in my stats book, *An introduction to Mathematical Statistics and Its Applications, *by Larsen and Marx. Alphonse Bertillon was a French criminologist who developed a system to identify criminals. He chose 11 anatomical characteristics that supposedly remain somewhat unchanged throughout a person’s adult life – such as ear length – and divided each characteristic into three classifications: small, medium, and large. So each of the 11 anatomical variables are categorized according to their respective size, and the ordered sequence of 11 letters (s for ‘small’, m for ‘medium’, and l for ‘large’) comprised a “Bertillon Configuration”.

This seems rudimentary and imprecise, perhaps, but keep in mind this was before the discovery of fingerprinting and so forth. One glaring issue with this system is the obvious subjectivity involved with assigning a classification to a given anatomical variable. Also, I’d imagine it’s kind of hard to get all of this information about a person, especially if they’re suspected of having committed a crime but haven’t ever been in custody.

Issues aside, the Bertillon configuration is an interesting idea I think. But the obvious question to ask is how likely is it that two people have the exact same Bertillon configuration? The implications are two people who, from a legal classification standpoint, are exactly the same (and you can use your imagination from there). So, how many people must live in a municipality before two of them necessarily share a Bertillon configuration?

The question is actually very simple to answer via the multiplication rule:

*If Operation A can be performed in m different ways, and Operation B can be performed in n different ways, the sequence {Operation A, Operation B} can be performed in (m)(n) different ways.*

*Corollary: If Operation A*_{i}*, i = 1, 2, …, k can be performed in n*_{i}* ways, i = 1, 2, …, k, respectively, then the ordered sequence, {Operation A*_{1}*, Operation A*_{2}*, …, Operation A*_{k}*) can be performed in (n*_{1}*)(n*_{2}*)***(n*_{k}*) ways.*

Think of Operation A as the anatomical variable and Operation B as assigning it a value (s,m,l).

Each Operation can be performed in three different ways, namely s, m, and l. Since there are 11 anatomical variables, by the multiplication rule there are (3)*(3)*(3)*(3)*(3)*(3)*(3)*(3) *(3)*(3)*(3) = 3^{11} ways to assign arrange the Bertillon configuration. Therefore, if 3^{11} = 177,147 people live in a town, we’d expect two of them to have the same Bertillon configuration.

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