Factorial of a Natural Number
It is the product of 
consecutive factors from to 
. The factorial of a number is denoted by 
.
Variations
Ordinary variations of 
elements taken at a time 
are defined as the different groups formed by elements such that:
- Not all elements are included
- Order matters
- Elements are not repeated
We can also calculate variations using factorials:
Variations are denoted by 
Variations with Repetition
Variations with repetition of elements taken at a time are defined as the different groups formed by elements such that:
- Not all elements are included if m > n. All elements may be included if
- Order matters
- Elements are repeated
Permutations
- All elements are included
- Order matters
- Elements are not repeated
Circular Permutations
They are used when elements must be arranged "in a circle" (for example, diners at a table), so that the first element "placed" in the sample determines the beginning and end of the sample.
Permutations with Repetition
Permutations with repetition of elements where the first element is repeated 
times, the second 
times, the third 
times,… such that 
, are the different groups that can be formed with those elements such that:
- All elements are included
- Order matters
- Elements are repeated
Combinations
Combinations of elements taken at a time are defined as all possible groupings that can be made with the elements such that:
- Not all elements are included
- Order does not matter
- Elements are not repeated
We can also calculate combinations using factorials:
Combinations with Repetition
Combinations with repetition of elements taken at a time are the different groups formed by elements such that:
- Not all elements are included
- Order does not matter
- Elements are repeated
Binomial Coefficients
The number 
is also called a binomial coefficient. It is represented by
and is read as "m choose n."
Properties of Binomial Coefficients
- 1.
- 2.
- 3.
Binomial Theorem
The formula that allows us to find the powers of a binomial is known as the Binomial Theorem.
If you wish to apply the theory with exercises on variations, combinations, and permutations, feel free to consult the other sections of this topic.
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