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Discrete Mathematics: Counting and Number Theory
A reviewed path from functions and finite counting to arrangements, selections, divisibility and the Euclidean algorithm. Every new concept is paired with verified video evidence, while the prerequisite arrows describe this map’s editorial learning order.
Functions and finite counting
Functions
A function assigns exactly one output to each input in its domain; its domain matters when comparing expressions.
2 reviewed resourcesCombinatorial counting
Count finite outcomes with disjoint-case sums, successive-choice products and equal-fiber division or bijections. Check that cases are disjoint and multiplicities are constant. Multiplication of choice counts is not a probability-independence assumption.
2 reviewed resources
Arrangements and selections
Permutations
Permutations record order. Distinct n objects have n! complete arrangements; indistinguishable repeated groups require dividing out their internal exchanges. Specify the multiplicities, whether all objects are used and whether rotations count as different arrangements.
2 reviewed resourcesCombinations
Choose k distinct elements from an n-element set without order, for integers 0≤k≤n. Each chosen subset corresponds to k!(n−k)! full permutations. Repetition-allowed selections are a different counting problem and need their own assumptions.
1 reviewed resources
Divisibility algorithms
Greatest common divisor
For integers a and b that are not both zero, the greatest common divisor is the unique positive integer dividing both that is divisible by every other common divisor. Signs do not change its positive value.
6 reviewed resourcesEuclidean algorithm
The Euclidean algorithm computes a greatest common divisor by repeated division with remainder. Replacing (a,b) by (b,r) preserves the common divisors, and the strictly smaller nonnegative remainders force termination.
5 reviewed resources