Combinatorics

It is the number of different ways in which we can order the elements of a set. It is calculated by:

It is read permutations of n elements.
Example: Calculate the number of words, with or without meaning, that can be formed with the letters MESA.

Permutations with repetition
The different ways in which n elements can be ordered, where one is repeated $a$ times, another is repeated $b$ times... etc.

Example: How many words, with or without meaning, can be formed with the letters CARCAJADA?
The set has 9 elements, of which the letter A is repeated 4 times and the letter C is repeated 2 times, then:

Given a set of m elements, the number of different ways in which n distinct elements can be ordered, are variations of m elements taken n at a time ($n \le m$)

Example: How many different 2-letter words can we form with the letters ABETO?

Variations with repetition
Given a set of m elements, the number of different sequences of n elements, including repetitions, that can be formed, are the variations with repetition of m elements taken n at a time

Example: How many 4-digit numbers can we form with the digits 1, 2, 3?
It is a set of 3 elements. We take them 4 at a time

Binomial coefficient
The binomial coefficient $\binom{m}{n}$, read $m$ choose $n$, is the operation:

Combinations
Given a set of m elements, the number of different subsets of n elements that we can extract are the combinations of m elements taken n at a time ($n \le m$)

Example: an exam consists of 10 questions from which 5 must be chosen. In how many different ways can we choose those 5 questions?

Combinations with repetition
Given m types of elements, the number of groups of n elements that we can form, including repetitions, are the combinations with repetition of m elements taken n at a time

Example: in a bakery there are 3 types of cakes. In how many different ways can we buy 5 cakes?