---
title: "Discrete Mathematics"
description: "12 topics in Mathematics, in the order they build on each other."
canonical: https://lightmysky.com/learn/mathematics/areas/discrete-mathematics
source: https://lightmysky.com/learn/mathematics/areas/discrete-mathematics.md
retrieved: 2026-09-02
---

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# Discrete Mathematics

12 topics in Mathematics, in the order they build on each other.

Page: https://lightmysky.com/learn/mathematics/areas/discrete-mathematics

- [Counting with the Product and Sum Rules](https://lightmysky.com/learn/mathematics/counting-with-the-product-and-sum-rules-mt_3NZC5F7RKP): Count a finite set by breaking the choice into independent stages to multiply, or into disjoint cases to add, and say which structure the problem has.
- [Permutations and Combinations](https://lightmysky.com/learn/mathematics/permutations-and-combinations-mt_1L58kuu-w3): Count arrangements where order matters and selections where it does not, and see the second as the first divided by the arrangements you refuse to distinguish.
- [Binomial Coefficients and Combinatorial Identities](https://lightmysky.com/learn/mathematics/binomial-coefficients-and-combinatorial-identities-mt_lHXFvudTh6): Read the binomial coefficient as a count and prove identities about it by counting one set two ways rather than by algebra.
- [The Pigeonhole Principle](https://lightmysky.com/learn/mathematics/the-pigeonhole-principle-mt_wgfpmZqzd9): Use the observation that more objects than boxes forces a repeat to prove existence results no construction would give.
- [Inclusion-Exclusion](https://lightmysky.com/learn/mathematics/inclusion-exclusion-mt_MgU-IMpVHA): Count a union by adding the parts, subtracting the pairwise overlaps, adding the triples back, and prove the alternating pattern is right.
- [Recurrence Relations and Characteristic Roots](https://lightmysky.com/learn/mathematics/recurrence-relations-and-characteristic-roots-mt_q0b2bi0iAB): Turn a rule that defines each term from earlier ones into a closed formula by solving the characteristic equation, including the repeated-root case.
- [Generating Functions](https://lightmysky.com/learn/mathematics/generating-functions-mt_WAX3O-HjRk): Store a whole sequence as the coefficients of a power series, then use algebra on the series to answer questions about the sequence.
- [Graphs, Degrees and the Handshake Lemma](https://lightmysky.com/learn/mathematics/graphs-degrees-and-the-handshake-lemma-mt_1oqO_o3bTO): Define a graph as a set of vertices with a set of pairs, and prove the first theorem about it: the degrees sum to twice the number of edges.
- [Trees and Spanning Trees](https://lightmysky.com/learn/mathematics/trees-and-spanning-trees-mt_kMWpsz44yJ): Characterise a tree in several equivalent ways, prove the edge count, and count the labelled trees on n vertices.
- [Euler Trails and Hamilton Cycles](https://lightmysky.com/learn/mathematics/euler-trails-and-hamilton-cycles-mt_RU5P5Iqekt): Settle exactly when a graph has a walk using every edge once, and see why the analogous question about vertices has no such clean answer.
- [Graph Colouring and Planarity](https://lightmysky.com/learn/mathematics/graph-colouring-and-planarity-mt_6xy6UPBppw): Colour vertices so that neighbours differ, bound the colours needed, and use Euler's formula to see what a graph drawn without crossings cannot contain.
- [Matchings and Hall's Theorem](https://lightmysky.com/learn/mathematics/matchings-and-halls-theorem-mt_hkWslW5b5U): Decide when every element of one side can be paired off, and prove that the obvious obstruction is the only one.