# Tagged Choice Counts and Actual Branch Widths

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261008-advance-20261012-r02-counting-principle-foundations · 2026-10-08 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(0, [])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (0, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0001/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 2. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[0]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(0, [0])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (0, [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0002/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 3. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[1]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(1, [1])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (1, [1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0003/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 4. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[2, 0, 3, 1]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(6, [2, 2, 5, 6])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (6, [2, 2, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0004/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 5. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[1, 1]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(2, [1, 2])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (2, [1, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0005/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 6. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[1, 2]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(3, [1, 3])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (3, [1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0006/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 7. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[1, 3]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(4, [1, 4])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (4, [1, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0007/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 8. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[1, 4]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(5, [1, 5])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (5, [1, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0008/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 9. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[1, 5]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(6, [1, 6])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (6, [1, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0009/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 10. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[1, 6]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(7, [1, 7])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (7, [1, 7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0010/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 11. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[1, 7]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(8, [1, 8])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (8, [1, 8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0011/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 12. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[1, 8]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(9, [1, 9])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (9, [1, 9]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0012/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 13. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[2, 1]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(3, [2, 3])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (3, [2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0013/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 14. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[2, 2]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(4, [2, 4])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (4, [2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0014/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 15. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[2, 3]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(5, [2, 5])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (5, [2, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0015/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 16. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[2, 4]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(6, [2, 6])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (6, [2, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0016/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 17. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[2, 5]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(7, [2, 7])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (7, [2, 7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0017/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 18. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[2, 6]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(8, [2, 8])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (8, [2, 8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0018/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 19. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[2, 7]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(9, [2, 9])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (9, [2, 9]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0019/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 20. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[2, 8]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(10, [2, 10])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (10, [2, 10]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0020/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 21. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[3, 1]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(4, [3, 4])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (4, [3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0021/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 22. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[3, 2]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(5, [3, 5])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (5, [3, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0022/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 23. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[3, 3]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(6, [3, 6])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (6, [3, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0023/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 24. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[3, 4]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(7, [3, 7])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (7, [3, 7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0024/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 25. Disjoint Tagged Choices and Prefix Counts

Mutually exclusive tagged categories have ordered sizes=[3, 5]. Give (total choices, each prefix’s total), retaining an empty list.

View answer and explanation(8, [3, 8])

Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (8, [3, 8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0025/)

Basic counting principles · qf-density12-counting-disjoint-choice-sizes · candidate





## 26. Full Product Choices and Prefix Counts

Ordered stage sizes=[]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(1, [])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (1, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0026/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 27. Full Product Choices and Prefix Counts

Ordered stage sizes=[0]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(0, [0])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (0, [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0027/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 28. Full Product Choices and Prefix Counts

Ordered stage sizes=[2, 0, 3]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(0, [2, 0, 0])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (0, [2, 0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0028/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 29. Full Product Choices and Prefix Counts

Ordered stage sizes=[0, 4, 5]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(0, [0, 0, 0])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (0, [0, 0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0029/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 30. Full Product Choices and Prefix Counts

Ordered stage sizes=[2, 3, 0, 8]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(0, [2, 6, 0, 0])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (0, [2, 6, 0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0030/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 31. Full Product Choices and Prefix Counts

Ordered stage sizes=[1, 1]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(1, [1, 1])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (1, [1, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0031/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 32. Full Product Choices and Prefix Counts

Ordered stage sizes=[1, 2]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(2, [1, 2])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (2, [1, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0032/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 33. Full Product Choices and Prefix Counts

Ordered stage sizes=[1, 3]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(3, [1, 3])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (3, [1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0033/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 34. Full Product Choices and Prefix Counts

Ordered stage sizes=[1, 4]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(4, [1, 4])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (4, [1, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0034/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 35. Full Product Choices and Prefix Counts

Ordered stage sizes=[1, 5]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(5, [1, 5])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (5, [1, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0035/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 36. Full Product Choices and Prefix Counts

Ordered stage sizes=[1, 6]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(6, [1, 6])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (6, [1, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0036/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 37. Full Product Choices and Prefix Counts

Ordered stage sizes=[1, 7]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(7, [1, 7])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (7, [1, 7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0037/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 38. Full Product Choices and Prefix Counts

Ordered stage sizes=[1, 8]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(8, [1, 8])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (8, [1, 8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0038/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 39. Full Product Choices and Prefix Counts

Ordered stage sizes=[2, 1]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(2, [2, 2])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (2, [2, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0039/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 40. Full Product Choices and Prefix Counts

Ordered stage sizes=[2, 2]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(4, [2, 4])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (4, [2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0040/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 41. Full Product Choices and Prefix Counts

Ordered stage sizes=[2, 3]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(6, [2, 6])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (6, [2, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0041/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 42. Full Product Choices and Prefix Counts

Ordered stage sizes=[2, 4]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(8, [2, 8])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (8, [2, 8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0042/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 43. Full Product Choices and Prefix Counts

Ordered stage sizes=[2, 5]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(10, [2, 10])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (10, [2, 10]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0043/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 44. Full Product Choices and Prefix Counts

Ordered stage sizes=[2, 6]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(12, [2, 12])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (12, [2, 12]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0044/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 45. Full Product Choices and Prefix Counts

Ordered stage sizes=[2, 7]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(14, [2, 14])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (14, [2, 14]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0045/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 46. Full Product Choices and Prefix Counts

Ordered stage sizes=[2, 8]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(16, [2, 16])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (16, [2, 16]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0046/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 47. Full Product Choices and Prefix Counts

Ordered stage sizes=[3, 1]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(3, [3, 3])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (3, [3, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0047/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 48. Full Product Choices and Prefix Counts

Ordered stage sizes=[3, 2]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(6, [3, 6])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (6, [3, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0048/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 49. Full Product Choices and Prefix Counts

Ordered stage sizes=[3, 3]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(9, [3, 9])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (9, [3, 9]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0049/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 50. Full Product Choices and Prefix Counts

Ordered stage sizes=[3, 4]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.

View answer and explanation(12, [3, 12])

Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (12, [3, 12]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0050/)

Basic counting principles · qf-density12-counting-full-product-prefix · candidate





## 51. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[], n=0; candidate common width h=7. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(0, 0, Yes, 0)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (0, 0, Yes, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0051/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 52. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[1, 3], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(4, 4, No, 0)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (4, 4, No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0052/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 53. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[2, 2], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(4, 4, Yes, 0)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (4, 4, Yes, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0053/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 54. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[0], n=1; candidate common width h=0. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(0, 0, Yes, 0)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (0, 0, Yes, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0054/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 55. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[3], n=1; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(3, 2, No, -1)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (3, 2, No, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0055/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 56. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[5, 6], n=2; candidate common width h=0. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(11, 0, No, -11)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (11, 0, No, -11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0056/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 57. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[6, 7], n=2; candidate common width h=0. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(13, 0, No, -13)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (13, 0, No, -13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0057/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 58. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[7, 8], n=2; candidate common width h=0. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(15, 0, No, -15)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (15, 0, No, -15).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0058/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 59. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[8, 0], n=2; candidate common width h=0. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(8, 0, No, -8)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (8, 0, No, -8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0059/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 60. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[0, 1], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(1, 2, No, 1)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (1, 2, No, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0060/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 61. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[1, 2], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(3, 2, No, -1)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (3, 2, No, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0061/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 62. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[2, 3], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(5, 2, No, -3)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (5, 2, No, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0062/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 63. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[3, 4], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(7, 2, No, -5)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (7, 2, No, -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0063/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 64. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[4, 5], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(9, 2, No, -7)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (9, 2, No, -7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0064/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 65. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[5, 6], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(11, 2, No, -9)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (11, 2, No, -9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0065/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 66. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[6, 7], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(13, 2, No, -11)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (13, 2, No, -11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0066/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 67. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[7, 8], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(15, 2, No, -13)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (15, 2, No, -13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0067/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 68. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[8, 0], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(8, 2, No, -6)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (8, 2, No, -6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0068/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 69. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[0, 1], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(1, 4, No, 3)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (1, 4, No, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0069/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 70. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[1, 2], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(3, 4, No, 1)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (3, 4, No, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0070/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 71. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[2, 3], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(5, 4, No, -1)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (5, 4, No, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0071/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 72. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[3, 4], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(7, 4, No, -3)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (7, 4, No, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0072/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 73. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[4, 5], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(9, 4, No, -5)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (9, 4, No, -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0073/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 74. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[5, 6], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(11, 4, No, -7)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (11, 4, No, -7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0074/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 75. Actual Branch Widths versus a Candidate Width

Ordered second-stage branch counts=[6, 7], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).

View answer and explanation(13, 4, No, -9)

Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (13, 4, No, -9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0075/)

Basic counting principles · qf-density12-counting-conditional-candidate-defect · candidate





## 76. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[], B=[]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(0, 0, 0)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (0, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0076/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 77. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0], B=[0]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(2, 1, 1)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (2, 1, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0077/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 78. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[], B=[100]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(1, 1, 0)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (1, 1, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0078/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 79. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0, 1], B=[0, 1]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(4, 2, 2)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (4, 2, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0079/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 80. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0], B=[0, 100]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(3, 2, 1)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (3, 2, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0080/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 81. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[], B=[100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(2, 2, 0)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (2, 2, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0081/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 82. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[-1], B=[100]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(2, 2, 0)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (2, 2, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0082/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 83. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0, 1, 2], B=[0, 1, 2]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(6, 3, 3)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (6, 3, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0083/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 84. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0, 1], B=[0, 1, 100]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(5, 3, 2)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (5, 3, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0084/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 85. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0], B=[0, 100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(4, 3, 1)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (4, 3, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0085/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 86. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[], B=[100, 101, 102]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(3, 3, 0)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (3, 3, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0086/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 87. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0, -1], B=[0, 100]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(4, 3, 1)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (4, 3, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0087/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 88. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[-1], B=[100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(3, 3, 0)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (3, 3, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0088/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 89. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0, 1, 2, 3], B=[0, 1, 2, 3]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(8, 4, 4)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (8, 4, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0089/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 90. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0, 1, 2], B=[0, 1, 2, 100]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(7, 4, 3)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (7, 4, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0090/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 91. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0, 1], B=[0, 1, 100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(6, 4, 2)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (6, 4, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0091/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 92. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0], B=[0, 100, 101, 102]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(5, 4, 1)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (5, 4, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0092/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 93. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[], B=[100, 101, 102, 103]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(4, 4, 0)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (4, 4, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0093/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 94. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0, 1, -1], B=[0, 1, 100]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(6, 4, 2)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (6, 4, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0094/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 95. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0, -1], B=[0, 100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(5, 4, 1)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (5, 4, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0095/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 96. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[-1], B=[100, 101, 102]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(4, 4, 0)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (4, 4, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0096/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 97. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[-2, -1], B=[100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(4, 4, 0)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (4, 4, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0097/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 98. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0, 1, 2, 3, 4], B=[0, 1, 2, 3, 4]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(10, 5, 5)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (10, 5, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0098/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 99. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0, 1, 2, 3], B=[0, 1, 2, 3, 100]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(9, 5, 4)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (9, 5, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0099/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate





## 100. Overlap-Corrected Finite Choice Counts

Finite integer choice sets A=[0, 1, 2], B=[0, 1, 2, 100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).

View answer and explanation(8, 5, 3)

Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (8, 5, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r02-counting-principle-0100/)

Basic counting principles · qf-density12-counting-overlap-correction · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261012-r02-counting-principle-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261008-advance-20261012-r02-counting-principle-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261012-r02-counting-principle-foundations/validation-report.json)