Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Discrete Mathematics ↓


Combinatorics


Combinatorics is a fascinating branch of mathematics that deals with counting, arranging and finding patterns in sets. It helps us understand the principles of permutation, combination and why certain results occur. In discrete mathematics, combinatorics is widely used to solve problems in computing, logic and probability. This study is essential to understand how to handle complex arrangements and sets efficiently.

Basic definitions and principles

At a basic level, combinatorics deals with two primary counting principles: permutations and combinations. These principles help determine the number of ways in which objects can be arranged or selected. Let's look at these concepts in more detail.

Permutation

Permutation involves arranging a group of objects in a specific order. The order of arrangement plays an important role here. For example, if you have three letters, say A, B, and C, and you want to arrange them in all possible ways, then you will be dealing with permutation.

The number of permutations of n distinct objects is given by n! (n factorial), which is the product of all positive integers up to n.

    n!= n × (n − 1) × (n − 2) × ... × 1

For example, the permutations of the set {A, B, C} are:

  • ABC
  • ACB
  • BAC
  • BCA
  • Cab
  • CBA

If n is 3 (the number of letters in the set), then the permutations are: 3! = 3 × 2 × 1 = 6.

Permutations of {A, B, C} {ABC, ACB, BAC, BCA, CAB, CBA}

Combination

Unlike permutations, combinations focus on choosing items from a set where the order does not matter. For example, when choosing 2 letters from the set {A, B, C}, the combinations are AB, AC, and BC. The formula for combinations is given as:

    C(n, r) = n! / (r! × (n - r)!)

Here, n is the total number of items, and r is the number of items to choose from.

For example, choosing 2 letters from a group of 3 letters means,

    c(3, 2) = 3! / (2! × (3 - 2)!) = 3
Combination of 2 from {A, B, C} {AB, AC, BC}

Advanced applications

In combinatorics, we also explore more complex operations involving repetitions, restrictions, and variation of objects.

Permutations with repetitions

Sometimes, you may have to arrange objects where repetition is allowed. If you have a set of n objects and you need to determine the number of ways to arrange them with repetition, you use the formula:

    n^r

Where n is the number of items to choose from, and r is the number you choose. For example, choosing 2 letters from {A, B} with repetitions:

    2^2 = 4 → AA, AB, BA, BB

Permutations with differentiable objects

Consider a scenario where you have indistinguishable items within a set, and you need to determine the number of distinct arrangements. For example, arranging the letters in the word "SASSY". Here, 'S' and 'S' are indistinguishable, just like any other pair of letters 'S' and 'S'.

    n! / (p! × q! × r!)

where p, q, r, ... are the frequencies of the indivisible elements. For "SASSY":

    5! / (3! × 1! × 1!) = 20

Combinations with repetition

In these scenarios, you select items from a set where repetition is allowed and the order does not matter. The formula used is:

    C(n + r − 1, r)

For example, in how many ways can you choose two fruits from apples and oranges if repetitions are allowed?

    c(2 + 2 - 1, 2) = c(3, 2) = 3

Combinatorial identities

Combinatorics also includes fascinating identities and theorems. One of the famous identities is the binomial theorem, which gives the expansion of (x + y)^n. The binomial theorem is expressed as:

    (x + y)^n = Σ C(n, k) * x^(nk) * y^k

This theorem reveals properties related to the coefficients called binomial coefficients and demonstrates the symmetry and pattern found in Pascal's triangle.

Let's consider an example for (x + y)^3:

    (x + y)^3 = C(3, 0)x^3 + C(3, 1)x^2y + C(3, 2)xy^2 + C(3, 3)y^3
              = 1*x^3 + 3*x^2y + 3*xy^2 + 1*y^3

Graph theory and combinatorics

Combinatorics is fundamental in graph theory, it is a field that deals with nodes and edges while finding efficient paths, maximum subgraphs and analyzing connectivity properties. Consider a basic example of finding the number of different paths in a grid.

Suppose you want to go from the top-left corner to the bottom-right corner using the shortest path. You have to find out how many such paths exist. This is a typical combinatorial exercise where you apply combinations to determine the number of ways to arrange movements.

Conclusion

Combinatorics is a wonderful field of mathematics that enables us to solve problems involving counting, arrangement, and selection. Its applications span across many fields, including computer science, logic, biology, and more. As you delve deeper into combinatorics, you will appreciate how beautifully it simplifies complex problems by breaking them down into manageable parts.


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