{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261109-r05-binomial-theorem-0068", "pilot_number": "P1000-0443", "number": 443, "family_id": "qf-mod40-binomial-evaluation-identity", "title": { "zh-Hant": "第 443 題：二項式定理", "en": "Question 443: Compare the Whole Binomial Sum with an Exact Integer Power" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261109-r05-binomial-theorem-0068", "batch_id": "batch-20261014-advance-20261109-r05-binomial-theorem-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-binomial-theorem" ], "title_zh": "二項式定理", "stem_zh": "commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":17,\"x\":0,\"y\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(17, 0, 0, 0, 0, 是)", "explanation_zh": "Wave6第三層核對=(17, 0, 0, 0, 0, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "二項式定理" ] }, "variant": { "family_id": "qf-mod40-binomial-evaluation-identity", "seed": 202611095068, "parameters": { "n": 17, "x": 0, "y": 0, "domain": "Exact_commutative_binomial_theorem", "operation": "binomial-evaluation-identity", "semantic_template": "qf-mod40-binomial-evaluation-identity/{\"n\":17,\"x\":0,\"y\":0}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261109-r05-binomial-theorem-0068", "batch_id": "batch-20261014-advance-20261109-r05-binomial-theorem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "b4d04cc67d8f4224f6aa02a5bca445c22897f247698da9a9655cf4e30da054c0", "family_id": "qf-mod40-binomial-evaluation-identity", "seed": 202611095068, "semantic_template": "qf-mod40-binomial-evaluation-identity/{\"n\":17,\"x\":0,\"y\":0}" }, "fields": { "title": "Compare the Whole Binomial Sum with an Exact Integer Power", "stem": "Use natural exponent n=17, exact commuting integers x=0, y=0. Give (n, x, y, power, expansion_sum, equal).", "answer": { "kind": "expression", "value": "(17, 0, 0, 0, 0, Yes)", "explanation": "Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (17, 0, 0, 0, 0, Yes)." }, "skills": [ "Compare the Whole Binomial Sum with an Exact Integer Power" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":17,\"x\":0,\"y\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use natural exponent n=17, exact commuting integers x=0, y=0. Give (n, x, y, power, expansion_sum, equal)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "n=17≥0; multiplication commutes", "after": "group the finite product by the number of y choices", "relation": "implies", "rule_id": "commutative_product", "conditions": [], "explanation": { "zh-Hant": "每個因子選x或y，依選y的次數分組，係數先作整數計數。", "en": "Choose x or y in each factor and group by the number of y choices; count coefficients as integers first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "x+y := sum(0, 0)", "after": "x+y = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先直接計算和。", "en": "First calculate the sum directly." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 x+y，不以小數近似替代。", "en": "Recompute x+y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "0" ], "output": "0", "label": "x+y" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "(x+y)^n: (0)^17", "after": "(x+y)^n = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "17" ], "output": "0", "label": "(x+y)^n" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "choose 0 positions out of 17", "after": "C(17,0)=355687428096000/(1·355687428096000)=1", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "0" ], "output": "1", "label": "C(17,0)" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "x^17: (0)^17", "after": "x^17 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "17" ], "output": "0", "label": "x^17" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "y^0: (0)^0", "after": "y^0 = 1", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "0" ], "output": "1", "label": "y^0" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "C·x power (k=0) := mul(1, 0)", "after": "C·x power (k=0) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=0)，不以小數近似替代。", "en": "Recompute C·x power (k=0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "0" ], "output": "0", "label": "C·x power (k=0)" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "term[0] := mul(0, 1)", "after": "term[0] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[0]，不以小數近似替代。", "en": "Recompute term[0] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "1" ], "output": "0", "label": "term[0]" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "choose 1 positions out of 17", "after": "C(17,1)=355687428096000/(1·20922789888000)=17", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "1" ], "output": "17", "label": "C(17,1)" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "x^16: (0)^16", "after": "x^16 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "16" ], "output": "0", "label": "x^16" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "y^1: (0)^1", "after": "y^1 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "1" ], "output": "0", "label": "y^1" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "C·x power (k=1) := mul(17, 0)", "after": "C·x power (k=1) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=1)，不以小數近似替代。", "en": "Recompute C·x power (k=1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "17", "0" ], "output": "0", "label": "C·x power (k=1)" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "term[1] := mul(0, 0)", "after": "term[1] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[1]，不以小數近似替代。", "en": "Recompute term[1] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[1]" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "choose 2 positions out of 17", "after": "C(17,2)=355687428096000/(2·1307674368000)=136", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "2" ], "output": "136", "label": "C(17,2)" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "x^15: (0)^15", "after": "x^15 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "15" ], "output": "0", "label": "x^15" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "y^2: (0)^2", "after": "y^2 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "2" ], "output": "0", "label": "y^2" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "C·x power (k=2) := mul(136, 0)", "after": "C·x power (k=2) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=2)，不以小數近似替代。", "en": "Recompute C·x power (k=2) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "136", "0" ], "output": "0", "label": "C·x power (k=2)" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "term[2] := mul(0, 0)", "after": "term[2] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[2]，不以小數近似替代。", "en": "Recompute term[2] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[2]" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "choose 3 positions out of 17", "after": "C(17,3)=355687428096000/(6·87178291200)=680", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "3" ], "output": "680", "label": "C(17,3)" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "x^14: (0)^14", "after": "x^14 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "14" ], "output": "0", "label": "x^14" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "y^3: (0)^3", "after": "y^3 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "3" ], "output": "0", "label": "y^3" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "C·x power (k=3) := mul(680, 0)", "after": "C·x power (k=3) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=3)，不以小數近似替代。", "en": "Recompute C·x power (k=3) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "680", "0" ], "output": "0", "label": "C·x power (k=3)" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "term[3] := mul(0, 0)", "after": "term[3] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[3]，不以小數近似替代。", "en": "Recompute term[3] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[3]" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "choose 4 positions out of 17", "after": "C(17,4)=355687428096000/(24·6227020800)=2380", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "4" ], "output": "2380", "label": "C(17,4)" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "x^13: (0)^13", "after": "x^13 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "13" ], "output": "0", "label": "x^13" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "y^4: (0)^4", "after": "y^4 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "4" ], "output": "0", "label": "y^4" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "C·x power (k=4) := mul(2380, 0)", "after": "C·x power (k=4) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=4)，不以小數近似替代。", "en": "Recompute C·x power (k=4) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2380", "0" ], "output": "0", "label": "C·x power (k=4)" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "term[4] := mul(0, 0)", "after": "term[4] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[4]，不以小數近似替代。", "en": "Recompute term[4] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[4]" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "choose 5 positions out of 17", "after": "C(17,5)=355687428096000/(120·479001600)=6188", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "5" ], "output": "6188", "label": "C(17,5)" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "x^12: (0)^12", "after": "x^12 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "12" ], "output": "0", "label": "x^12" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "y^5: (0)^5", "after": "y^5 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "5" ], "output": "0", "label": "y^5" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "C·x power (k=5) := mul(6188, 0)", "after": "C·x power (k=5) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=5)，不以小數近似替代。", "en": "Recompute C·x power (k=5) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6188", "0" ], "output": "0", "label": "C·x power (k=5)" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "term[5] := mul(0, 0)", "after": "term[5] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[5]，不以小數近似替代。", "en": "Recompute term[5] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[5]" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "choose 6 positions out of 17", "after": "C(17,6)=355687428096000/(720·39916800)=12376", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "6" ], "output": "12376", "label": "C(17,6)" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "x^11: (0)^11", "after": "x^11 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "11" ], "output": "0", "label": "x^11" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "y^6: (0)^6", "after": "y^6 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "6" ], "output": "0", "label": "y^6" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "C·x power (k=6) := mul(12376, 0)", "after": "C·x power (k=6) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=6)，不以小數近似替代。", "en": "Recompute C·x power (k=6) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "12376", "0" ], "output": "0", "label": "C·x power (k=6)" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "term[6] := mul(0, 0)", "after": "term[6] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[6]，不以小數近似替代。", "en": "Recompute term[6] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[6]" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "choose 7 positions out of 17", "after": "C(17,7)=355687428096000/(5040·3628800)=19448", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "7" ], "output": "19448", "label": "C(17,7)" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "x^10: (0)^10", "after": "x^10 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "10" ], "output": "0", "label": "x^10" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "y^7: (0)^7", "after": "y^7 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "7" ], "output": "0", "label": "y^7" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "C·x power (k=7) := mul(19448, 0)", "after": "C·x power (k=7) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=7)，不以小數近似替代。", "en": "Recompute C·x power (k=7) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "19448", "0" ], "output": "0", "label": "C·x power (k=7)" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "term[7] := mul(0, 0)", "after": "term[7] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[7]，不以小數近似替代。", "en": "Recompute term[7] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[7]" } }, { "id": "S044", "uses": [ "G001", "S043" ], "before": "choose 8 positions out of 17", "after": "C(17,8)=355687428096000/(40320·362880)=24310", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "8" ], "output": "24310", "label": "C(17,8)" } }, { "id": "S045", "uses": [ "G001", "S044" ], "before": "x^9: (0)^9", "after": "x^9 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "9" ], "output": "0", "label": "x^9" } }, { "id": "S046", "uses": [ "G001", "S045" ], "before": "y^8: (0)^8", "after": "y^8 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "8" ], "output": "0", "label": "y^8" } }, { "id": "S047", "uses": [ "G001", "S046" ], "before": "C·x power (k=8) := mul(24310, 0)", "after": "C·x power (k=8) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=8)，不以小數近似替代。", "en": "Recompute C·x power (k=8) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24310", "0" ], "output": "0", "label": "C·x power (k=8)" } }, { "id": "S048", "uses": [ "G001", "S047" ], "before": "term[8] := mul(0, 0)", "after": "term[8] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[8]，不以小數近似替代。", "en": "Recompute term[8] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[8]" } }, { "id": "S049", "uses": [ "G001", "S048" ], "before": "choose 9 positions out of 17", "after": "C(17,9)=355687428096000/(362880·40320)=24310", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "9" ], "output": "24310", "label": "C(17,9)" } }, { "id": "S050", "uses": [ "G001", "S049" ], "before": "x^8: (0)^8", "after": "x^8 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "8" ], "output": "0", "label": "x^8" } }, { "id": "S051", "uses": [ "G001", "S050" ], "before": "y^9: (0)^9", "after": "y^9 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "9" ], "output": "0", "label": "y^9" } }, { "id": "S052", "uses": [ "G001", "S051" ], "before": "C·x power (k=9) := mul(24310, 0)", "after": "C·x power (k=9) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=9)，不以小數近似替代。", "en": "Recompute C·x power (k=9) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24310", "0" ], "output": "0", "label": "C·x power (k=9)" } }, { "id": "S053", "uses": [ "G001", "S052" ], "before": "term[9] := mul(0, 0)", "after": "term[9] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[9]，不以小數近似替代。", "en": "Recompute term[9] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[9]" } }, { "id": "S054", "uses": [ "G001", "S053" ], "before": "choose 10 positions out of 17", "after": "C(17,10)=355687428096000/(3628800·5040)=19448", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "10" ], "output": "19448", "label": "C(17,10)" } }, { "id": "S055", "uses": [ "G001", "S054" ], "before": "x^7: (0)^7", "after": "x^7 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "7" ], "output": "0", "label": "x^7" } }, { "id": "S056", "uses": [ "G001", "S055" ], "before": "y^10: (0)^10", "after": "y^10 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "10" ], "output": "0", "label": "y^10" } }, { "id": "S057", "uses": [ "G001", "S056" ], "before": "C·x power (k=10) := mul(19448, 0)", "after": "C·x power (k=10) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=10)，不以小數近似替代。", "en": "Recompute C·x power (k=10) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "19448", "0" ], "output": "0", "label": "C·x power (k=10)" } }, { "id": "S058", "uses": [ "G001", "S057" ], "before": "term[10] := mul(0, 0)", "after": "term[10] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[10]，不以小數近似替代。", "en": "Recompute term[10] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[10]" } }, { "id": "S059", "uses": [ "G001", "S058" ], "before": "choose 11 positions out of 17", "after": "C(17,11)=355687428096000/(39916800·720)=12376", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "11" ], "output": "12376", "label": "C(17,11)" } }, { "id": "S060", "uses": [ "G001", "S059" ], "before": "x^6: (0)^6", "after": "x^6 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "6" ], "output": "0", "label": "x^6" } }, { "id": "S061", "uses": [ "G001", "S060" ], "before": "y^11: (0)^11", "after": "y^11 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "11" ], "output": "0", "label": "y^11" } }, { "id": "S062", "uses": [ "G001", "S061" ], "before": "C·x power (k=11) := mul(12376, 0)", "after": "C·x power (k=11) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=11)，不以小數近似替代。", "en": "Recompute C·x power (k=11) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "12376", "0" ], "output": "0", "label": "C·x power (k=11)" } }, { "id": "S063", "uses": [ "G001", "S062" ], "before": "term[11] := mul(0, 0)", "after": "term[11] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[11]，不以小數近似替代。", "en": "Recompute term[11] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[11]" } }, { "id": "S064", "uses": [ "G001", "S063" ], "before": "choose 12 positions out of 17", "after": "C(17,12)=355687428096000/(479001600·120)=6188", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "12" ], "output": "6188", "label": "C(17,12)" } }, { "id": "S065", "uses": [ "G001", "S064" ], "before": "x^5: (0)^5", "after": "x^5 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "5" ], "output": "0", "label": "x^5" } }, { "id": "S066", "uses": [ "G001", "S065" ], "before": "y^12: (0)^12", "after": "y^12 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "12" ], "output": "0", "label": "y^12" } }, { "id": "S067", "uses": [ "G001", "S066" ], "before": "C·x power (k=12) := mul(6188, 0)", "after": "C·x power (k=12) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=12)，不以小數近似替代。", "en": "Recompute C·x power (k=12) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6188", "0" ], "output": "0", "label": "C·x power (k=12)" } }, { "id": "S068", "uses": [ "G001", "S067" ], "before": "term[12] := mul(0, 0)", "after": "term[12] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[12]，不以小數近似替代。", "en": "Recompute term[12] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[12]" } }, { "id": "S069", "uses": [ "G001", "S068" ], "before": "choose 13 positions out of 17", "after": "C(17,13)=355687428096000/(6227020800·24)=2380", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "13" ], "output": "2380", "label": "C(17,13)" } }, { "id": "S070", "uses": [ "G001", "S069" ], "before": "x^4: (0)^4", "after": "x^4 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "4" ], "output": "0", "label": "x^4" } }, { "id": "S071", "uses": [ "G001", "S070" ], "before": "y^13: (0)^13", "after": "y^13 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "13" ], "output": "0", "label": "y^13" } }, { "id": "S072", "uses": [ "G001", "S071" ], "before": "C·x power (k=13) := mul(2380, 0)", "after": "C·x power (k=13) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=13)，不以小數近似替代。", "en": "Recompute C·x power (k=13) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2380", "0" ], "output": "0", "label": "C·x power (k=13)" } }, { "id": "S073", "uses": [ "G001", "S072" ], "before": "term[13] := mul(0, 0)", "after": "term[13] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[13]，不以小數近似替代。", "en": "Recompute term[13] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[13]" } }, { "id": "S074", "uses": [ "G001", "S073" ], "before": "choose 14 positions out of 17", "after": "C(17,14)=355687428096000/(87178291200·6)=680", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "14" ], "output": "680", "label": "C(17,14)" } }, { "id": "S075", "uses": [ "G001", "S074" ], "before": "x^3: (0)^3", "after": "x^3 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "3" ], "output": "0", "label": "x^3" } }, { "id": "S076", "uses": [ "G001", "S075" ], "before": "y^14: (0)^14", "after": "y^14 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "14" ], "output": "0", "label": "y^14" } }, { "id": "S077", "uses": [ "G001", "S076" ], "before": "C·x power (k=14) := mul(680, 0)", "after": "C·x power (k=14) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=14)，不以小數近似替代。", "en": "Recompute C·x power (k=14) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "680", "0" ], "output": "0", "label": "C·x power (k=14)" } }, { "id": "S078", "uses": [ "G001", "S077" ], "before": "term[14] := mul(0, 0)", "after": "term[14] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[14]，不以小數近似替代。", "en": "Recompute term[14] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[14]" } }, { "id": "S079", "uses": [ "G001", "S078" ], "before": "choose 15 positions out of 17", "after": "C(17,15)=355687428096000/(1307674368000·2)=136", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "15" ], "output": "136", "label": "C(17,15)" } }, { "id": "S080", "uses": [ "G001", "S079" ], "before": "x^2: (0)^2", "after": "x^2 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "2" ], "output": "0", "label": "x^2" } }, { "id": "S081", "uses": [ "G001", "S080" ], "before": "y^15: (0)^15", "after": "y^15 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "15" ], "output": "0", "label": "y^15" } }, { "id": "S082", "uses": [ "G001", "S081" ], "before": "C·x power (k=15) := mul(136, 0)", "after": "C·x power (k=15) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=15)，不以小數近似替代。", "en": "Recompute C·x power (k=15) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "136", "0" ], "output": "0", "label": "C·x power (k=15)" } }, { "id": "S083", "uses": [ "G001", "S082" ], "before": "term[15] := mul(0, 0)", "after": "term[15] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[15]，不以小數近似替代。", "en": "Recompute term[15] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[15]" } }, { "id": "S084", "uses": [ "G001", "S083" ], "before": "choose 16 positions out of 17", "after": "C(17,16)=355687428096000/(20922789888000·1)=17", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "16" ], "output": "17", "label": "C(17,16)" } }, { "id": "S085", "uses": [ "G001", "S084" ], "before": "x^1: (0)^1", "after": "x^1 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "1" ], "output": "0", "label": "x^1" } }, { "id": "S086", "uses": [ "G001", "S085" ], "before": "y^16: (0)^16", "after": "y^16 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "16" ], "output": "0", "label": "y^16" } }, { "id": "S087", "uses": [ "G001", "S086" ], "before": "C·x power (k=16) := mul(17, 0)", "after": "C·x power (k=16) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=16)，不以小數近似替代。", "en": "Recompute C·x power (k=16) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "17", "0" ], "output": "0", "label": "C·x power (k=16)" } }, { "id": "S088", "uses": [ "G001", "S087" ], "before": "term[16] := mul(0, 0)", "after": "term[16] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[16]，不以小數近似替代。", "en": "Recompute term[16] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "term[16]" } }, { "id": "S089", "uses": [ "G001", "S088" ], "before": "choose 17 positions out of 17", "after": "C(17,17)=355687428096000/(355687428096000·1)=1", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "17", "17" ], "output": "1", "label": "C(17,17)" } }, { "id": "S090", "uses": [ "G001", "S089" ], "before": "x^0: (0)^0", "after": "x^0 = 1", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "0" ], "output": "1", "label": "x^0" } }, { "id": "S091", "uses": [ "G001", "S090" ], "before": "y^17: (0)^17", "after": "y^17 = 0", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次的有限乘積計算；零次為乘法單位元。", "en": "Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "0", "17" ], "output": "0", "label": "y^17" } }, { "id": "S092", "uses": [ "G001", "S091" ], "before": "C·x power (k=17) := mul(1, 1)", "after": "C·x power (k=17) = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把係數乘上x項。", "en": "Multiply the coefficient by the x power first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 C·x power (k=17)，不以小數近似替代。", "en": "Recompute C·x power (k=17) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "C·x power (k=17)" } }, { "id": "S093", "uses": [ "G001", "S092" ], "before": "term[17] := mul(1, 0)", "after": "term[17] = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘y項，保留負底數的整個括號。", "en": "Multiply by the y power, preserving parentheses around a negative base." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 term[17]，不以小數近似替代。", "en": "Recompute term[17] with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "0" ], "output": "0", "label": "term[17]" } }, { "id": "S094", "uses": [ "G001", "S093" ], "before": "expanded sum := sum(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)", "after": "expanded sum = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "所有k都已列出，逐項加總，不只取首尾項。", "en": "Sum every displayed k term, not merely the endpoints." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 expanded sum，不以小數近似替代。", "en": "Recompute expanded sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "output": "0", "label": "expanded sum" } }, { "id": "S095", "uses": [ "G001", "S094" ], "before": "direct=0; expanded=0", "after": "identity holds: 是", "relation": "implies", "rule_id": "finite_expansion_evaluation", "conditions": [], "explanation": { "zh-Hant": "兩條計算路徑一致，核對本題的有限二項恆等式。", "en": "The two calculation paths agree on this finite instance of the identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": null } ], "math_model": { "result": [ 17, 0, 0, "0", "0", true ] }, "conclusion": { "canonical_value": "(17, 0, 0, 0, 0, 是)", "explanation": { "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。", "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings." } }, "definition_note": { "zh-Hant": "n 為非負整數且 x、y 可交換；逐項 C(n,k)x^(n−k)y^k 加總，零次冪為 1。", "en": "n is a nonnegative integer and x,y commute. Sum every term C(n,k)x^(n−k)y^k; the zeroth power is 1." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261109-r05-binomial-theorem-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261109-r05-binomial-theorem-foundations/questions.jsonl", "batch_sha256": "f83302c1353681ecb1209ee1ccbef0d266f15fb9415191ed9289eb329e1175cc", "original_line": 68, "original_line_sha256": "f94f4b14981bdcda9a832b0a5251993ec2e1d27a64746c54684755e763f8d1ee", "question_compact_sha256": "b4d04cc67d8f4224f6aa02a5bca445c22897f247698da9a9655cf4e30da054c0", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261109-r05-binomial-theorem-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261109-r05-binomial-theorem-foundations.en.jsonl", "english_row_compact_sha256": "465c5ff197dae9957b14a114e2d8e18770bd39eb9482f66c15ef4581de11ae07" }, "mko_bindings": { "mko-binomial-theorem": { "sha256": "c8fefe3783fc4f7633b4b22b8cf149d96a683900f181e3b42c690409ce90e5b7", "english_sha256": "1086dfe7c31f4e6bf8934681a0a080b50c14cb828f1a8b17e59143ec2cf06c83", "original_path": "public/data/mko/binomial-theorem.json", "english_path": "public/data/i18n/editions/mko-binomial-theorem.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "afa7c0e9f7084cc5929124dcffe9929d5db7def32e40047056d19ea25535873e", "original_candidate_sha256": "b1d502f8f5155b79c3b2c8500a048839baa85d82174ca760b91adceba9c40e4f", "original_candidate_line_sha256": "b1d502f8f5155b79c3b2c8500a048839baa85d82174ca760b91adceba9c40e4f" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。", "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI review or general formal proof. Original questions, answers, MKOs, formulas, ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-003-01", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "Thirty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct sentence shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full answer tuples, model results and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772", "question_first_witness_sha256": "77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 95, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "12d99ca4c6648ceff51a67365d305e824465a7ae1ae68e2596a71e7fb7cfcb01", "arithmetic_steps": 93 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }