Combinatorics

The math behind variations, permutations, and combinations.

Variations With Repetition

Variations are the total number of ways to pick and arrange some elements of a given set

For example, a lock with 2 letter code, each can be A, B, or C. If A was chosen for 1st letter, 3 variations exist for the second letter. Total variations are 3 * 3 = 9.

$$\bar{V^{n}_{p}} = n^{p}$$

The number of variations with repetition when picking p-many elements out of n elements, is equal to n to the power of p.

Variations Without Repetition

Once an element has been chosen, it cannot be used or considered again in future calculations.

A simple formula to calculate permutations for n slots for n elements.

$$n!$$

Formal notation and formula for the scenario:

$$\bar{V^{n}_{p}} = \frac{n!}{(n-p)!}$$

The number of variations without repetition when arranging p elements out of a total of n.

Combinations

Order in which elements are chosen are not relevant, just that each element is different.

We can say that all the different permutations of a single combination are different variations.

$$C^{n}_{p} = \frac{n!}{p!(n-p)!}$$

The number of combinations for choosing p-many elements out of a sample space of n elements. The number of combinations equals the number of variations over the number of permutations.

Combination Symmetry

The more chosen elements, the lesser the combinations. We can pick p-many elements in as many ways as we can pick n - p many elements.

$$C^{n}_{p} = C^{n}_{n-p}$$

Combinations With Separate Sample Spaces

E.g. You are at a restaurant, need to buy a sandwich (3 choices), a side (2 choices), and a drink (2 choices). Think of each section (sandwich, side, drink), together, and multiply the choices available for each.

$$3 * 2 * 2 = 12$$