Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Discrete Mathematics → Combinatorics ↓


Permutations and Combinations


In the exciting world of discrete mathematics, two fundamental concepts - permutations and combinations - play a vital role in understanding how we can count and arrange the various elements in a set. These basic ideas not only come in handy in pure mathematical theories, but also find wide applications in statistics, computer science, cryptography, and other fields.

Understanding the basics

Before diving deeper into permutations and combinations, let's define what each one intuitively means:

  • Permutation refers to the different ways in which objects or numbers can be arranged or ordered.
  • Combinations are different ways of selecting items from a set, where the order does not matter.

To understand these concepts, let us understand each one step by step and go through definitions, formulas, and examples to see how they differ.

Permutation

A permutation is an arrangement of all or part of a group of objects, with respect to the order of the arrangement. The simplest example of a permutation is the arrangement of numbers. For example, given the numbers 1, 2 and 3, the permutations of these three could be: 123, 132, 213, 231, 312 and 321. Note that changing the order produces a different arrangement, hence a different permutation.

Mathematical expression of permutation

Generally, if we have a set of n different items, and we want to know how many ways we can arrange r of the n items, we use permutations. The formula for permutations, usually denoted as P(n, r) or nPr, is given as:

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

where n! (n factorial) is the product of all positive integers up to n.

Example of permutation

Let us consider an example to understand this concept better:

Suppose we want to know in how many ways we can arrange 3 out of 5 books on a shelf. Here, n = 5 and r = 3 Using the permutation formula:

P(5, 3) = 5! / (5-3)! = 5! / 2! = (5 x 4 x 3 x 2 x 1) / (2 x 1) = 60

Thus, there are 60 different ways to arrange 3 of 5 books on a shelf.

Visual example of permutation

To see this, consider the set of letters {A, B, C}. The permutations of this set are:

ABC ACB BAC BCA CAB CBA

As seen, each arrangement is different because of the importance of order in the permutation.

Combination

In combination, attention is paid to the selection of objects, not to their order. This means that in combination, attention is paid only to the presence of objects, not to their arrangement.

Mathematical expression of combinations

The combination formula, traditionally denoted C(n, r) or nCr, is:

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

This formula shows the number of ways to select r elements from a group of n elements, regardless of the order of selection.

Example of combinations

For example, consider how many ways you can pick 3 fruits from a basket containing 5 fruits (apples, bananas, cherries, dates, and elderberries). This means n = 5 and r = 3 applying the formula:

C(5, 3) = 5! / (3! x (5-3)!) = 5! / (3! x 2!) = (5 x 4 x 3 x 2 x 1) / (3 x 2 x 1 x 2 x 1) = 10

Hence, there are 10 different ways to choose 3 fruits from a group of 5 fruits.

Visual example of combinations

To explain further, let's use the set of letters {A, B, C} and pick 2 letters:

AB AC BC

Note here that combinations do not consider the order of selection, so AB is the same as BA.

Main differences between permutation and combination

Although permutations and combinations may seem similar, they fundamentally differ based on whether the order matters or not:

  • In permutation, different arrangements are considered and the order in which the elements are selected is important.
  • Combination focuses only on the selection of items, not the order.

To see this important difference, the number of permutations for the same values of n and r is usually greater than the number of combinations.

Applications of permutations and combinations

These fundamental concepts of combinatorics are not just theoretical; they also have practical applications in many real-world areas:

  • Cryptography: Permutations are essential in creating complex encryption algorithms, which ensure data security by arranging characters in a particular sequence.
  • Statistics: Combinations are important in designing studies and experiments when selecting sample groups from a larger population.
  • Computer science: Both permutations and combinations are used in algorithms related to searching and sorting techniques.

Advanced topics in permutation and combination

For undergraduate students, permutations and combinations are just a gateway to deeper topics of combinatorics. Here are some advanced aspects that can be understood in more detail:

  • Restricted permutations: Exploring scenarios where certain elements should always or never appear in specific positions.
  • Combinatorial recognizers: Understanding Binomial Coefficients and Pascal's Triangle for Complex Calculations.
  • Generating functions: using algebraic expressions that encode information about sequences and can help solve combinatorial problems.

Conclusion

In conclusion, permutations and combinations provide a fundamental framework for counting and arranging objects in discrete mathematics. The practical applications and theoretical depth of these concepts make them essential tools in a variety of mathematical and applied fields. By exploring and mastering these ideas, you can better understand the beauty and utility of mathematics in understanding the complex systems of the world.


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