Alphabeta Math

Finite Counting, Factorials and Binomial Coefficients: Examples and Counterexamples

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How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 14 results · all verified · 8 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 6 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Combinatorics
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

Pascal's triangle computed to row 6, with Pascal's rule checked at every interior entry

Example

Choosing a committee: C(10, 3) = 120, and the ordered count 10^3̲ = 720

Example

Arrangements of a word with repeated letters, counted by the multinomial coefficient

Example

Five-card hands from a 52-card deck: C(52, 5) = 2598960, and the count of hands with all cards of one suit

Example

The 15 weak compositions of 4 into 3 parts, listed and matched against stars and bars

Example

All 8 functions 3 → 2, the 0 injections 3 → 2, and the 8 subsets of a 3-element set

Example

Vandermonde's identity checked at m = 4, n = 3, k = 3, both sides equal to 35

Counterexample

Two sets of the same finite cardinality between which the bijection is not unique

Counterexample

A count that overcounts because the blocks are not disjoint, and exactly where the sum rule's hypothesis is spent

False statement

FALSE: every injection of a set into itself is a bijection

False statement

FALSE: | A ∪ B| = | A| + | B| for all finite A and B

False statement

FALSE: for all sets A and B with B having at least two elements, A × B is strictly larger than A

False statement

FALSE: ∑_k<n+1(-1)ᵏiotaC(n, k) = 0 for every n ∈ ℕ

False statement

FALSE: the number of weak compositions of n into m parts is C(n+m-1, m-1) for every m ∈ ℕ

Sources