PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Combinatorics → Enumerative Combinatorics ↓


Permutations and Combinations


Permutations and combinations are fundamental concepts in the world of mathematics, especially in the field of combinatorics. They help us understand how to count and organize data logically. These concepts are particularly useful for solving problems where we need to determine all possible arrangements or selections of a given item.

Let’s dig deeper into these concepts, starting with the basic understanding before moving on to more complex scenarios.

Basic definitions

Permutation

Permutation is arranging elements in a specific order. The order in permutation is important because changing the order of the elements will result in a different permutation. For example, consider a set of letters {A, B, C}. The different permutations of these letters are as follows:

ABC, ACB, BAC, BCA, CAB, CBA

The arrangement "BAC" is different from "ACB" because the order of the letters is changed.

Combination

Combination refers to the selection of elements without considering the order. Only the selection of elements matters, not their order. For example, using the same set of letters {A, B, C}, the different combinations of two letters are as follows:

AB, AC, BC

Here, "AB" is the same as "BA", because the order in the combinations does not matter.

Mathematical representation of permutations and combinations

Permutation formula

The formula to count the number of permutations of 'n' distinct elements by taking 'r' at a time is as follows:

P(n, r) = n! / (n - r)!

Where 'n!' (n factorial) represents the product of all positive integers up to 'n'. For example, if you have 5 different books and you want to arrange 3 of them on a shelf, you can calculate the number of permutations as follows:

P(5, 3) = 5! / (5-3)! = 5 × 4 × 3 = 60

Combination formulas

The formula to count the combinations of 'n' different elements by taking 'r' at a time is as follows:

C(n, r) = n! / (r! × (n - r)!)

Using the same book example, if you want to choose 3 books from a total of 5 without considering the order, the number of combinations will be:

C(5, 3) = 5! / (3! × (5-3)!) = 10

Visual example

Permutations of {A, B, C}

ABC ACB BAC BCA Cab CBA

Combination of {A, B, C} (choose 2)

Now AC BC

Examples in real-world scenarios

Permutations in real life

  • Seating arrangement: Imagine you are hosting a dinner party and you have 8 guests. You want to know how many different ways you can arrange them around a round table. This is a permutation problem because the order in which the guests are seated matters.
  • Password and code: Suppose you are creating a 4-digit numeric password. If we use the digits 0 to 9, the number of possible passwords is a permutation of 10 digits, 4 of which are taken at a time.

Combinations in real life

  • Lottery games: In many lottery games players choose a set of numbers. The order does not matter, so this is a combination problem.
  • Selection of committee: If you have a club with 10 members and you need to select a committee of 3 members, this is a combination problem, because the order of selection does not matter.

More advanced topics

Permutations with repetitions

In some problems, we may have to take into account the repetition of elements. The formula for permutation with repetition is:

n^r

Where 'n' is the number of things to choose from, and 'r' is the number of things you want to choose from. For example, if you have 3 different ice cream flavors and you want to make a 2-scoop cone where the flavors can be repeated, you calculate it like this:

3^2 = 9

Different permutations would include: vanilla-vanilla, vanilla-chocolate, vanilla-strawberry, etc.

Combinations with repetition

In combinations, if repetition is allowed, the situation becomes a bit more complicated. The formula for combinations with repetition is given as follows:

C(n + r - 1, r) = (n + r - 1)! / (r! × (n - 1)!)

Suppose you are choosing 3 scoops of ice cream from 3 flavors, and the flavors can be repeated. The number of combinations is:

C(3 + 3 - 1, 3) = 10

Applications in computer science

Permutations and combinations are not just abstract mathematical concepts; they are essential to algorithms and data structures in computer science. For example:

  • Search algorithms: Finding different paths or sequences often involves generating permutations or combinations of steps.
  • Cryptography: Secure systems rely on the vast number of permutations available to securely encrypt data.

Conclusion

Understanding permutations and combinations is important for anyone entering mathematics, computer science, or any field requiring analytical problem-solving skills. These concepts enable us to solve complex problems diligently and thoughtfully. Through practice and application in real-world scenarios, one can appreciate their versatility and important role in analytical topics.


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Permutations and Combinations

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