What are the applications of permutations of a finite set?

[dwn5000]

I am having trouble understanding the permutations of a finite set in general. I want to know what it may be used for, and how to solve some of its problems (examples?). In my attachment, I post some pictures of what I am currently reading, and what has confused me.

 

[homeomorphic]

Permutations are useful for counting things. For example, you might use them to calculate probabilities. Calculating probabilities is of great practical importance.

 

[chiro]

Permutations apart from probability are also important for linear algebra and tensor theory.

Geometry itself has important attributes regarding permutations that relate to the characterization of orientation. The determinant itself which is a signed measure has an important permutation property relating to how the determinant is not only calculated, but also derived.

In tensor theory, there is a special permutation tensor (which I think is called the Christoffel symbol) that is one foundation for understanding permutations and can be related back to the ideas of linear algebra.

The other important connection is to discrete mathematics especially for graphs and algorithms (apart from probability and general counting).

You can indirectly relate it back to number theory and other periodic processes (like waves) if you want as well.

 

[marschmellow]

Permutations apart from probability are also important for linear algebra and tensor theory.

Geometry itself has important attributes regarding permutations that relate to the characterization of orientation. The determinant itself which is a signed measure has an important permutation property relating to how the determinant is not only calculated, but also derived.

In tensor theory, there is a special permutation tensor (which I think is called the Christoffel symbol) that is one foundation for understanding permutations and can be related back to the ideas of linear algebra.

The other important connection is to discrete mathematics especially for graphs and algorithms (apart from probability and general counting).

You can indirectly relate it back to number theory and other periodic processes (like waves) if you want as well.

I believe it's the Levi-Civita symbol, and it can be made into a tensor density.

Permutations will show up in the most random of places, including in real life. It's probably been the so-far most useful topic I learned in algebra.