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-20260914-domain-foundations · 2026-09-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
In the number system N, including 0, how should the result of 2+3 be classified?
5
2+3=5, which remains in the explicitly stated natural-number system N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 0+8 be classified?
8
0+8=8, which remains in the explicitly stated natural-number system N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 11+7 be classified?
18
11+7=18, which remains in the explicitly stated natural-number system N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 24+16 be classified?
40
24+16=40, which remains in the explicitly stated natural-number system N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 100+9 be classified?
109
100+9=109, which remains in the explicitly stated natural-number system N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 2*4 be classified?
8
2*4=8, which remains in the explicitly stated natural-number system N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 0*15 be classified?
0
0*15=0, which remains in the explicitly stated natural-number system N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 3*7 be classified?
21
3*7=21, which remains in the explicitly stated natural-number system N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 8*9 be classified?
72
8*9=72, which remains in the explicitly stated natural-number system N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 12*5 be classified?
60
12*5=60, which remains in the explicitly stated natural-number system N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 3-5 be classified?
Not closed in N
3-5 is not a natural number in this instance, so the operation is not closed in N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 0-1 be classified?
Not closed in N
0-1 is not a natural number in this instance, so the operation is not closed in N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 7-10 be classified?
Not closed in N
7-10 is not a natural number in this instance, so the operation is not closed in N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 14-20 be classified?
Not closed in N
14-20 is not a natural number in this instance, so the operation is not closed in N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 2-9 be classified?
Not closed in N
2-9 is not a natural number in this instance, so the operation is not closed in N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 3/2 be classified?
Not closed in N
3/2 is not a natural number in this instance, so the operation is not closed in N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 5/4 be classified?
Not closed in N
5/4 is not a natural number in this instance, so the operation is not closed in N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 7/3 be classified?
Not closed in N
7/3 is not a natural number in this instance, so the operation is not closed in N.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 9/0 be classified?
Zero divisor; no ordinary numerical result
The divisor in 9/0 is zero, so this question does not treat it as an ordinary numerical division result.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In the number system N, including 0, how should the result of 0/0 be classified?
Zero divisor; no ordinary numerical result
The divisor in 0/0 is zero, so this question does not treat it as an ordinary numerical division result.
Reviewed step-by-step derivation
Arithmetic operations · qf-arithmetic-domain-boundary · candidate
In N, let x=0. Evaluate the algebraic expression x+1; this is an expression, not an equation.
1
Substitution gives 0+1=1; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=3. Evaluate the algebraic expression x+2; this is an expression, not an equation.
5
Substitution gives 3+2=5; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=6. Evaluate the algebraic expression x+3; this is an expression, not an equation.
9
Substitution gives 6+3=9; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=9. Evaluate the algebraic expression x+4; this is an expression, not an equation.
13
Substitution gives 9+4=13; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=12. Evaluate the algebraic expression x+5; this is an expression, not an equation.
17
Substitution gives 12+5=17; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=15. Evaluate the algebraic expression x+1; this is an expression, not an equation.
16
Substitution gives 15+1=16; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=18. Evaluate the algebraic expression x+2; this is an expression, not an equation.
20
Substitution gives 18+2=20; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=21. Evaluate the algebraic expression x+3; this is an expression, not an equation.
24
Substitution gives 21+3=24; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=24. Evaluate the algebraic expression x+4; this is an expression, not an equation.
28
Substitution gives 24+4=28; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=27. Evaluate the algebraic expression x+5; this is an expression, not an equation.
32
Substitution gives 27+5=32; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=30. Evaluate the algebraic expression x+1; this is an expression, not an equation.
31
Substitution gives 30+1=31; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=33. Evaluate the algebraic expression x+2; this is an expression, not an equation.
35
Substitution gives 33+2=35; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=36. Evaluate the algebraic expression x+3; this is an expression, not an equation.
39
Substitution gives 36+3=39; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=39. Evaluate the algebraic expression x+4; this is an expression, not an equation.
43
Substitution gives 39+4=43; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=42. Evaluate the algebraic expression x+5; this is an expression, not an equation.
47
Substitution gives 42+5=47; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=45. Evaluate the algebraic expression x+1; this is an expression, not an equation.
46
Substitution gives 45+1=46; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=48. Evaluate the algebraic expression x+2; this is an expression, not an equation.
50
Substitution gives 48+2=50; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=51. Evaluate the algebraic expression x+3; this is an expression, not an equation.
54
Substitution gives 51+3=54; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=54. Evaluate the algebraic expression x+4; this is an expression, not an equation.
58
Substitution gives 54+4=58; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
In N, let x=57. Evaluate the algebraic expression x+5; this is an expression, not an equation.
62
Substitution gives 57+5=62; only the explicitly stated addition is used.
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-variable-expression-evaluation · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 1, label 2. At this observation scale the candidate sample space is S={label 1, label 2}. Is it complete?
Yes
The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 11, label 12, label 13. At this observation scale the candidate sample space is S={label 11, label 13}. Is it complete?
No
The candidate omits at least one explicitly possible elementary outcome, so it is incomplete.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 21, label 22, label 23, label 24. At this observation scale the candidate sample space is S={label 21, label 22, label 23, label 24, outcome outside this experiment 3}. Is it complete?
No
The candidate includes an outcome outside the explicitly stated experiment, so it is not the exact sample space.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 31, label 32, label 33, label 34, label 35. At this observation scale the candidate sample space is S={label 31, label 32, label 33, label 34, label 35}. Is it complete?
Yes
The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 41, label 42. At this observation scale the candidate sample space is S={label 42}. Is it complete?
No
The candidate omits at least one explicitly possible elementary outcome, so it is incomplete.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 51, label 52, label 53. At this observation scale the candidate sample space is S={label 51, label 52, label 53, outcome outside this experiment 6}. Is it complete?
No
The candidate includes an outcome outside the explicitly stated experiment, so it is not the exact sample space.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 61, label 62, label 63, label 64. At this observation scale the candidate sample space is S={label 61, label 62, label 63, label 64}. Is it complete?
Yes
The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 71, label 72, label 73, label 74, label 75. At this observation scale the candidate sample space is S={label 71, label 72, label 74, label 75}. Is it complete?
No
The candidate omits at least one explicitly possible elementary outcome, so it is incomplete.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 81, label 82. At this observation scale the candidate sample space is S={label 81, label 82, outcome outside this experiment 9}. Is it complete?
No
The candidate includes an outcome outside the explicitly stated experiment, so it is not the exact sample space.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 91, label 92, label 93. At this observation scale the candidate sample space is S={label 91, label 92, label 93}. Is it complete?
Yes
The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 101, label 102, label 103, label 104. At this observation scale the candidate sample space is S={label 101, label 102, label 104}. Is it complete?
No
The candidate omits at least one explicitly possible elementary outcome, so it is incomplete.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 111, label 112, label 113, label 114, label 115. At this observation scale the candidate sample space is S={label 111, label 112, label 113, label 114, label 115, outcome outside this experiment 12}. Is it complete?
No
The candidate includes an outcome outside the explicitly stated experiment, so it is not the exact sample space.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 121, label 122. At this observation scale the candidate sample space is S={label 121, label 122}. Is it complete?
Yes
The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 131, label 132, label 133. At this observation scale the candidate sample space is S={label 131, label 133}. Is it complete?
No
The candidate omits at least one explicitly possible elementary outcome, so it is incomplete.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 141, label 142, label 143, label 144. At this observation scale the candidate sample space is S={label 141, label 142, label 143, label 144, outcome outside this experiment 15}. Is it complete?
No
The candidate includes an outcome outside the explicitly stated experiment, so it is not the exact sample space.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 151, label 152, label 153, label 154, label 155. At this observation scale the candidate sample space is S={label 151, label 152, label 153, label 154, label 155}. Is it complete?
Yes
The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 161, label 162. At this observation scale the candidate sample space is S={label 162}. Is it complete?
No
The candidate omits at least one explicitly possible elementary outcome, so it is incomplete.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 171, label 172, label 173. At this observation scale the candidate sample space is S={label 171, label 172, label 173, outcome outside this experiment 18}. Is it complete?
No
The candidate includes an outcome outside the explicitly stated experiment, so it is not the exact sample space.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 181, label 182, label 183, label 184. At this observation scale the candidate sample space is S={label 181, label 182, label 183, label 184}. Is it complete?
Yes
The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
The experiment is one draw from explicitly labelled cards with elementary outcomes label 191, label 192, label 193, label 194, label 195. At this observation scale the candidate sample space is S={label 191, label 192, label 193, label 194}. Is it complete?
No
The candidate omits at least one explicitly possible elementary outcome, so it is incomplete.
Reviewed step-by-step derivation
Sample space · qf-finite-sample-space-completeness · candidate
For a proposition family P(n) beginning at 0, a draft supplies induction hypothesis, induction step, conclusion and checks 1 finite example(s). Which induction role is missing?
base case
The missing role is the base case. Finite examples do not discharge that proof obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For induction on P(n) from 0, which proof role is represented by the supplied clause “base case”?
base case
The clause has the role of the base case; each induction role has a distinct scope and obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For a proposition family P(n) beginning at 0, a draft supplies base case, induction step, conclusion and checks 3 finite example(s). Which induction role is missing?
induction hypothesis
The missing role is the induction hypothesis. Finite examples do not discharge that proof obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For induction on P(n) from 0, which proof role is represented by the supplied clause “induction hypothesis”?
induction hypothesis
The clause has the role of the induction hypothesis; each induction role has a distinct scope and obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For a proposition family P(n) beginning at 0, a draft supplies base case, induction hypothesis, conclusion and checks 5 finite example(s). Which induction role is missing?
induction step
The missing role is the induction step. Finite examples do not discharge that proof obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For induction on P(n) from 0, which proof role is represented by the supplied clause “induction step”?
induction step
The clause has the role of the induction step; each induction role has a distinct scope and obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For a proposition family P(n) beginning at 0, a draft supplies base case, induction hypothesis, induction step and checks 7 finite example(s). Which induction role is missing?
conclusion
The missing role is the conclusion. Finite examples do not discharge that proof obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For induction on P(n) from 0, which proof role is represented by the supplied clause “conclusion”?
conclusion
The clause has the role of the conclusion; each induction role has a distinct scope and obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For a proposition family P(n) beginning at 0, a draft supplies induction hypothesis, induction step, conclusion and checks 9 finite example(s). Which induction role is missing?
base case
The missing role is the base case. Finite examples do not discharge that proof obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For induction on P(n) from 0, which proof role is represented by the supplied clause “base case”?
base case
The clause has the role of the base case; each induction role has a distinct scope and obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For a proposition family P(n) beginning at 0, a draft supplies base case, induction step, conclusion and checks 11 finite example(s). Which induction role is missing?
induction hypothesis
The missing role is the induction hypothesis. Finite examples do not discharge that proof obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For induction on P(n) from 0, which proof role is represented by the supplied clause “induction hypothesis”?
induction hypothesis
The clause has the role of the induction hypothesis; each induction role has a distinct scope and obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For a proposition family P(n) beginning at 0, a draft supplies base case, induction hypothesis, conclusion and checks 13 finite example(s). Which induction role is missing?
induction step
The missing role is the induction step. Finite examples do not discharge that proof obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For induction on P(n) from 0, which proof role is represented by the supplied clause “induction step”?
induction step
The clause has the role of the induction step; each induction role has a distinct scope and obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For a proposition family P(n) beginning at 0, a draft supplies base case, induction hypothesis, induction step and checks 15 finite example(s). Which induction role is missing?
conclusion
The missing role is the conclusion. Finite examples do not discharge that proof obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For induction on P(n) from 0, which proof role is represented by the supplied clause “conclusion”?
conclusion
The clause has the role of the conclusion; each induction role has a distinct scope and obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For a proposition family P(n) beginning at 0, a draft supplies induction hypothesis, induction step, conclusion and checks 17 finite example(s). Which induction role is missing?
base case
The missing role is the base case. Finite examples do not discharge that proof obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For induction on P(n) from 0, which proof role is represented by the supplied clause “base case”?
base case
The clause has the role of the base case; each induction role has a distinct scope and obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For a proposition family P(n) beginning at 0, a draft supplies base case, induction step, conclusion and checks 19 finite example(s). Which induction role is missing?
induction hypothesis
The missing role is the induction hypothesis. Finite examples do not discharge that proof obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
For induction on P(n) from 0, which proof role is represented by the supplied clause “induction hypothesis”?
induction hypothesis
The clause has the role of the induction hypothesis; each induction role has a distinct scope and obligation.
Reviewed step-by-step derivation
Mathematical induction · qf-induction-proof-obligation · candidate
A task can be completed through two explicitly disjoint branches with 2 and 3 outcomes. What is the total?
5
The branches are explicitly disjoint, so 2+3=5.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A process has two successive stages. The first has 3 choices, and every first-stage choice is followed by exactly 5 second-stage choices. How many combinations are there?
15
Every first-stage choice has a fixed 5 continuations, so 3×5=15.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
Two branches report 4 and 5 outcomes, but both include shared outcome 3. May their counts be added unconditionally?
Cannot add unconditionally
The shared outcome belongs to both branches, so direct addition would count it twice.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A process has 5 first-stage choices, but the number of second-stage options depends on the first choice and can be 4 or 5. May one fixed product be used unconditionally?
Cannot multiply unconditionally
The second-stage count changes with the first choice, so one unconditional fixed product is invalid.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A task can be completed through two explicitly disjoint branches with 6 and 7 outcomes. What is the total?
13
The branches are explicitly disjoint, so 6+7=13.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A process has two successive stages. The first has 3 choices, and every first-stage choice is followed by exactly 3 second-stage choices. How many combinations are there?
9
Every first-stage choice has a fixed 3 continuations, so 3×3=9.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
Two branches report 8 and 3 outcomes, but both include shared outcome 7. May their counts be added unconditionally?
Cannot add unconditionally
The shared outcome belongs to both branches, so direct addition would count it twice.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A process has 5 first-stage choices, but the number of second-stage options depends on the first choice and can be 7 or 8. May one fixed product be used unconditionally?
Cannot multiply unconditionally
The second-stage count changes with the first choice, so one unconditional fixed product is invalid.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A task can be completed through two explicitly disjoint branches with 10 and 5 outcomes. What is the total?
15
The branches are explicitly disjoint, so 10+5=15.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A process has two successive stages. The first has 3 choices, and every first-stage choice is followed by exactly 6 second-stage choices. How many combinations are there?
18
Every first-stage choice has a fixed 6 continuations, so 3×6=18.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
Two branches report 12 and 7 outcomes, but both include shared outcome 11. May their counts be added unconditionally?
Cannot add unconditionally
The shared outcome belongs to both branches, so direct addition would count it twice.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A process has 5 first-stage choices, but the number of second-stage options depends on the first choice and can be 5 or 6. May one fixed product be used unconditionally?
Cannot multiply unconditionally
The second-stage count changes with the first choice, so one unconditional fixed product is invalid.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A task can be completed through two explicitly disjoint branches with 14 and 3 outcomes. What is the total?
17
The branches are explicitly disjoint, so 14+3=17.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A process has two successive stages. The first has 3 choices, and every first-stage choice is followed by exactly 4 second-stage choices. How many combinations are there?
12
Every first-stage choice has a fixed 4 continuations, so 3×4=12.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
Two branches report 16 and 5 outcomes, but both include shared outcome 15. May their counts be added unconditionally?
Cannot add unconditionally
The shared outcome belongs to both branches, so direct addition would count it twice.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A process has 5 first-stage choices, but the number of second-stage options depends on the first choice and can be 3 or 4. May one fixed product be used unconditionally?
Cannot multiply unconditionally
The second-stage count changes with the first choice, so one unconditional fixed product is invalid.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A task can be completed through two explicitly disjoint branches with 18 and 7 outcomes. What is the total?
25
The branches are explicitly disjoint, so 18+7=25.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A process has two successive stages. The first has 3 choices, and every first-stage choice is followed by exactly 7 second-stage choices. How many combinations are there?
21
Every first-stage choice has a fixed 7 continuations, so 3×7=21.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
Two branches report 20 and 3 outcomes, but both include shared outcome 19. May their counts be added unconditionally?
Cannot add unconditionally
The shared outcome belongs to both branches, so direct addition would count it twice.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
A process has 5 first-stage choices, but the number of second-stage options depends on the first choice and can be 6 or 7. May one fixed product be used unconditionally?
Cannot multiply unconditionally
The second-stage count changes with the first choice, so one unconditional fixed product is invalid.
Reviewed step-by-step derivation
Basic counting principles · qf-basic-counting-principle · candidate
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