Combinatorics -> Counting Basics¶

Factorials, binomials, arrangements, stars and bars, and the basic models that recur across counting problems.

Microtopics¶

Source	Type
MIT Mathematics for Computer Science	`Reference`
cp-algorithms binomial coefficients	`Reference`
OI Wiki combinations	`Reference`

Source	Type
CSES Binomial Coefficients	`Practice`
CSES Distributing Apples	`Practice`

Source	Type
USACO Guide Catalan Numbers	`Reference`

Problem	Source	Difficulty	Context	Style	Prerequisites	Tags	Why it fits
Bit Strings	`CSES`	`Easy`	Basic Counting, Mod Arithmetic	-	-	Powers Of Two	The simplest counting-by-choices starter.
Choose Cards 1	`AtCoder`	`Easy`	Pair-Counting	Implementation; Counting	Basic Counting; Frequency Arrays	Pairs	A very direct counting warm-up with only frequency bookkeeping.
Choose Cards 3	`AtCoder`	`Easy`	Pair-Counting	Implementation; Counting	Frequency Arrays; Complement Counting	Two-Sum; Frequency Counting	A great basic complement-counting exercise with a fixed target sum.
Combination Easy	`AtCoder`	`Easy`	Binomial-Coefficients	Math	Factorials; Combinations	Ncr	A pure combinatorics starter that isolates nCr without any hidden twists.
Two Knights	`CSES`	`Easy`	-	-	-	Board Counting; Pair Counting; Formula	Classic counting-by-cases on a board.
Binomial Coefficients	`CSES`	`Medium`	Modular-Arithmetic	Math; Precomputation	Factorials; Modular Inverse	Ncr; Factorials; Modular Inverse	The standard modular-combinatorics benchmark for factorial precomputation.
Choose Cards 2	`AtCoder`	`Medium`	Subset-Counting	Search; Counting	Subset Enumeration; Sum Constraints	Meet-In-The-Middle	A classic counting-by-subsets problem that sits between brute force and combinatorics.
Creating Strings II	`CSES`	`Medium`	Multiset Permutations	Math; Precomputation	Multiset Counting; Modular Arithmetic	Factorials	A textbook multinomial-counting problem with duplicate letters.
Two Sets II	`CSES`	`Medium`	DP	-	-	Partitions; Subset Sum; Mod Arithmetic	Nice introductory partition-counting problem.

Problem	Source	Difficulty	Context	Style	Prerequisites	Tags	Why it fits
Christmas Party	`CSES`	`Medium`	Derangements	Combinatorics; DP	Basic Counting; Inclusion-Exclusion	Derangements; Inclusion-Exclusion	A classic derangement/counting problem that is good practice for constrained permutations.
Distributing Apples	`CSES`	`Medium`	Stars-And-Bars	Math; Combinatorics	Combinations	Compositions; Combinations; Mod Arithmetic	A canonical stars-and-bars problem and a clean step toward distributions with constraints.

Problem	Source	Difficulty	Context	Style	Prerequisites	Tags	Why it fits
Counting Necklaces	`CSES`	`Hard`	Group-Actions	Combinatorics	Modular Arithmetic	Burnside; Pólya	A more advanced counting benchmark that moves from raw counting into symmetry.
Counting Permutations	`CSES`	`Hard`	Permutation-Counting, Inclusion-Style Counting	DP; Combinatorics	Inclusion-Exclusion Mindset	Permutations; Adjacency Constraints; Adjacency Restriction; Beautiful Permutations	A stronger permutation-counting problem with a clean but nontrivial constraint.

Code	Title	Fit	Difficulty	Pattern	Note	Solution
`DISTRIBUTINGAPPLES`	Distributing Apples	`primary`	`medium`	stars and bars; single binomial query; factorial precompute	Note	Code
`VOSFENCE`	Xay hang rao	`secondary`	`hard`	bounded compositions; run decomposition; gap distribution	Note	Code

python3 scripts/generate_problem_catalog.py