As cameldrv said, information theory. The late David MacKay has a good free book:

http://www.inference.org.uk/itprnn/book.html

The intro chapters cover the topic of measuring informational entropy.

Aside: One interesting application is to the problem of “use a balance scale only three times to determine which of twelve balls has the wrong weight (too heavy or light)”. You choose the weighings so as as to maximize the entropy of the outcome i.e. so there are many possible outcomes of equal probability.

Not the person you asked but

4! is the number of permutations of a list - it's commonly seen throughout CS theory and other forms of discrete math.

log_2(n) is the number of bits needed to store a integer from 0 to n - it's reasonably commonly used in CS theory education.

As others have mentioned, it comes up in information theory, or maybe in some subfields of combinatorics and probability. I learned it tangentially in various parts of physics.

But I might just recommend going to the source! which is Claude Shannon's seminal masters thesis on coding: http://affect-reason-utility.com/1301/4/shannon1948.pdf . It's surprisingly readable and worth at least skimming if you like this stuff.

try combinatorics branch of mathematics

Probability theory and information theory.