Number of Variations: The Cornerstone of Combinatorial Thinking
Apr 10, 2025
In the finals of the eSport World Championship, there are 8 competitors. In how many ways can medals be distributed if we know that the top three finishers receive medals?
For this type of problem, it matters who stands on the podium and in what order they finish.
Selection: who stands on the podium. Order: what place they finish in.
Let's create a model: 1st place: 8 possibilities 2nd place: 7 possibilities 3rd place: 6 possibilities
Therefore, the number of possibilities: 8×7×6 = 336.
Historical Background
The theory of calculating variations is an important chapter in mathematics history from the 17th-18th centuries. The work of Blaise Pascal and Jacob Bernoulli established the systematic foundation of combinatorics. Bernoulli's "Ars Conjectandi" (1713) already discussed variations and their applications in detail, which also laid the groundwork for probability theory.
Expressing the problem generally:
In how many ways can we select k objects from n different objects when the order of selection matters (k≤n)?
Definition:
When we create sequences of length k (k≤n) from elements of a set with n elements, where the order is important and each element can only be selected once, we call this process variation.
The resulting sequences (a specific selection in a specific arrangement) are called variations without repetition. The total number of possibilities, the number of variations of n elements of class k, is denoted by V(n,k).
Variation therefore involves both selection and order.
Theorem:
The number of variations of class k from "n" distinct elements: V(n,k) = n!/(n−k)!
Proof:
1st position: n possibilities
2nd position: (n-1) possibilities
3rd position: (n-2) possibilities
(k-1)th position: n-k+2 possibilities
kth position: n-k+1 possibilities
The total number of possibilities is given by the product of possibilities for each position: V(n,k) = n(n-1)(n-2)…(n-k+2)(n-k+1)
This is a product of k factors, multiplying the positive integers from n down to n-k+1.
Let's transform the expression by continuing the product all the way down to 1, meaning we multiply by (n-k)(n-k-1)(n-k-2)…3⋅2⋅1. To keep the value unchanged, we must also divide by these same factors.
The product (n-k)(n-k-1)(n-k-2)…3⋅2⋅1 equals (n-k)!. With this operation, we get the factorial of n (n!) when calculating permutations.
Thus, for the number of variations of class k from n elements, we get: V(n,k) = n!/(n−k)!, where k≤n.
Practical Applications
Calculating variations is essential in many real-world problems:
Drawings and Competitions: Figuring out possible podium positions in sports competitions (like the Olympics).
Genetics and Bioinformatics: Analyzing possible variations of nucleotide sequences in DNA. For example, the number of possible variations of DNA segments of length k from n nucleotides is V(n,k).
Cryptography: Calculating possible combinations of passwords and encryption keys. The number of passwords of length k from an alphabet of n characters (for variation with repetition) is n^k.
Logistics and Production Planning: Optimizing the order of manufacturing and shipping products. If n tasks need to be performed on k machines in a specific order, the possible arrangements can be calculated using variations.
Why Is This Important for Students?
Variations aren't just theoretical concepts but tools for developing critical thinking. When you learn to calculate variations:
You develop organizational skills
You learn to think systematically about complex problems
You acquire skills fundamental to computer science, data science, economics, and other fields
You gain the basics of algorithmic thinking, which gives you a head start when learning programming
Generally:
If we create sequences of length k from elements of a set with n elements where order matters and elements can be selected multiple times, we're talking about variations with repetition.
Theorem:
The number of variations with repetition of "n" elements with "k" members is n^k. That is: V(n,k,i) = n^k.
Connections to Other Fields
Computer Science and Data Science: In analyzing algorithm complexity, the number of possible execution paths is often determined using variation calculations. It's also essential in optimizing database queries.
Genetics: In calculating genetic variations and analyzing DNA sequences. Different sequences of the four nucleotides (A, C, G, T) in human DNA determine genetic information.
Economics: In decision theory, when the order of different factors influences the outcome. For example, analyzing different market entry strategies executed in various orders.
Linguistics: In examining possible word formations and sentence structures. The number of words of length k that can be created with an alphabet of n letters (variation with repetition).
Variation calculation is therefore not just a mathematical tool but a universal method for scientific and practical problem-solving that helps organize our thinking when analyzing complex possibilities.
In the finals of the eSport World Championship, there are 8 competitors. In how many ways can medals be distributed if we know that the top three finishers receive medals?
For this type of problem, it matters who stands on the podium and in what order they finish.
Selection: who stands on the podium. Order: what place they finish in.
Let's create a model: 1st place: 8 possibilities 2nd place: 7 possibilities 3rd place: 6 possibilities
Therefore, the number of possibilities: 8×7×6 = 336.
Historical Background
The theory of calculating variations is an important chapter in mathematics history from the 17th-18th centuries. The work of Blaise Pascal and Jacob Bernoulli established the systematic foundation of combinatorics. Bernoulli's "Ars Conjectandi" (1713) already discussed variations and their applications in detail, which also laid the groundwork for probability theory.
Expressing the problem generally:
In how many ways can we select k objects from n different objects when the order of selection matters (k≤n)?
Definition:
When we create sequences of length k (k≤n) from elements of a set with n elements, where the order is important and each element can only be selected once, we call this process variation.
The resulting sequences (a specific selection in a specific arrangement) are called variations without repetition. The total number of possibilities, the number of variations of n elements of class k, is denoted by V(n,k).
Variation therefore involves both selection and order.
Theorem:
The number of variations of class k from "n" distinct elements: V(n,k) = n!/(n−k)!
Proof:
1st position: n possibilities
2nd position: (n-1) possibilities
3rd position: (n-2) possibilities
(k-1)th position: n-k+2 possibilities
kth position: n-k+1 possibilities
The total number of possibilities is given by the product of possibilities for each position: V(n,k) = n(n-1)(n-2)…(n-k+2)(n-k+1)
This is a product of k factors, multiplying the positive integers from n down to n-k+1.
Let's transform the expression by continuing the product all the way down to 1, meaning we multiply by (n-k)(n-k-1)(n-k-2)…3⋅2⋅1. To keep the value unchanged, we must also divide by these same factors.
The product (n-k)(n-k-1)(n-k-2)…3⋅2⋅1 equals (n-k)!. With this operation, we get the factorial of n (n!) when calculating permutations.
Thus, for the number of variations of class k from n elements, we get: V(n,k) = n!/(n−k)!, where k≤n.
Practical Applications
Calculating variations is essential in many real-world problems:
Drawings and Competitions: Figuring out possible podium positions in sports competitions (like the Olympics).
Genetics and Bioinformatics: Analyzing possible variations of nucleotide sequences in DNA. For example, the number of possible variations of DNA segments of length k from n nucleotides is V(n,k).
Cryptography: Calculating possible combinations of passwords and encryption keys. The number of passwords of length k from an alphabet of n characters (for variation with repetition) is n^k.
Logistics and Production Planning: Optimizing the order of manufacturing and shipping products. If n tasks need to be performed on k machines in a specific order, the possible arrangements can be calculated using variations.
Why Is This Important for Students?
Variations aren't just theoretical concepts but tools for developing critical thinking. When you learn to calculate variations:
You develop organizational skills
You learn to think systematically about complex problems
You acquire skills fundamental to computer science, data science, economics, and other fields
You gain the basics of algorithmic thinking, which gives you a head start when learning programming
Generally:
If we create sequences of length k from elements of a set with n elements where order matters and elements can be selected multiple times, we're talking about variations with repetition.
Theorem:
The number of variations with repetition of "n" elements with "k" members is n^k. That is: V(n,k,i) = n^k.
Connections to Other Fields
Computer Science and Data Science: In analyzing algorithm complexity, the number of possible execution paths is often determined using variation calculations. It's also essential in optimizing database queries.
Genetics: In calculating genetic variations and analyzing DNA sequences. Different sequences of the four nucleotides (A, C, G, T) in human DNA determine genetic information.
Economics: In decision theory, when the order of different factors influences the outcome. For example, analyzing different market entry strategies executed in various orders.
Linguistics: In examining possible word formations and sentence structures. The number of words of length k that can be created with an alphabet of n letters (variation with repetition).
Variation calculation is therefore not just a mathematical tool but a universal method for scientific and practical problem-solving that helps organize our thinking when analyzing complex possibilities.
