# Question 433: Compare the Whole Binomial Sum with an Exact Integer Power [Return to catalogue page 3](https://ocme.evemisslab.com/en/questions/page/3/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261109-r05-binomial-theorem-foundations/#q-20261014-advance-20261109-r05-binomial-theorem-0058) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. P1000-0433 · q-20261014-advance-20261109-r05-binomial-theorem-0058 ## Original question and givens Use natural exponent n=7, exact commuting integers x=-3, y=0. Give (n, x, y, power, expansion_sum, equal). - G001 — Use natural exponent n=7, exact commuting integers x=-3, y=0. Give (n, x, y, power, expansion_sum, equal). ## Definition and scope 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. - [mko-binomial-theorem](https://ocme.evemisslab.com/en/mko/mko-binomial-theorem/) ## Step-by-step derivation S001 · impliesBefore n=7≥0; multiplication commutes After group the finite product by the number of y choices Conditions: none Choose x or y in each factor and group by the number of y choices; count coefficients as integers first. Step identifiers and dependenciesRule: commutative_product · Uses: G001 S002 · calculationBefore x+y := sum(-3, 0) After x+y = -3 Conditions: none First calculate the sum directly. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore (x+y)^n: (-3)^7 After (x+y)^n = -2187 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S002 S004 · impliesBefore choose 0 positions out of 7 After C(7,0)=5040/(1·5040)=1 Conditions: none Count each k-position subset once, dividing out reorderings inside each of the two groups. Step identifiers and dependenciesRule: subset_count · Uses: G001, S003 S005 · impliesBefore x^7: (-3)^7 After x^7 = -2187 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S004 S006 · impliesBefore y^0: (0)^0 After y^0 = 1 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S005 S007 · calculationBefore C·x power (k=0) := mul(1, -2187) After C·x power (k=0) = -2187 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore term[0] := mul(-2187, 1) After term[0] = -2187 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore choose 1 positions out of 7 After C(7,1)=5040/(1·720)=7 Conditions: none Count each k-position subset once, dividing out reorderings inside each of the two groups. Step identifiers and dependenciesRule: subset_count · Uses: G001, S008 S010 · impliesBefore x^6: (-3)^6 After x^6 = 729 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S009 S011 · impliesBefore y^1: (0)^1 After y^1 = 0 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S010 S012 · calculationBefore C·x power (k=1) := mul(7, 729) After C·x power (k=1) = 5103 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore term[1] := mul(5103, 0) After term[1] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore choose 2 positions out of 7 After C(7,2)=5040/(2·120)=21 Conditions: none Count each k-position subset once, dividing out reorderings inside each of the two groups. Step identifiers and dependenciesRule: subset_count · Uses: G001, S013 S015 · impliesBefore x^5: (-3)^5 After x^5 = -243 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S014 S016 · impliesBefore y^2: (0)^2 After y^2 = 0 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S015 S017 · calculationBefore C·x power (k=2) := mul(21, -243) After C·x power (k=2) = -5103 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore term[2] := mul(-5103, 0) After term[2] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore choose 3 positions out of 7 After C(7,3)=5040/(6·24)=35 Conditions: none Count each k-position subset once, dividing out reorderings inside each of the two groups. Step identifiers and dependenciesRule: subset_count · Uses: G001, S018 S020 · impliesBefore x^4: (-3)^4 After x^4 = 81 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S019 S021 · impliesBefore y^3: (0)^3 After y^3 = 0 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S020 S022 · calculationBefore C·x power (k=3) := mul(35, 81) After C·x power (k=3) = 2835 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore term[3] := mul(2835, 0) After term[3] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore choose 4 positions out of 7 After C(7,4)=5040/(24·6)=35 Conditions: none Count each k-position subset once, dividing out reorderings inside each of the two groups. Step identifiers and dependenciesRule: subset_count · Uses: G001, S023 S025 · impliesBefore x^3: (-3)^3 After x^3 = -27 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S024 S026 · impliesBefore y^4: (0)^4 After y^4 = 0 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S025 S027 · calculationBefore C·x power (k=4) := mul(35, -27) After C·x power (k=4) = -945 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore term[4] := mul(-945, 0) After term[4] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore choose 5 positions out of 7 After C(7,5)=5040/(120·2)=21 Conditions: none Count each k-position subset once, dividing out reorderings inside each of the two groups. Step identifiers and dependenciesRule: subset_count · Uses: G001, S028 S030 · impliesBefore x^2: (-3)^2 After x^2 = 9 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S029 S031 · impliesBefore y^5: (0)^5 After y^5 = 0 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S030 S032 · calculationBefore C·x power (k=5) := mul(21, 9) After C·x power (k=5) = 189 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore term[5] := mul(189, 0) After term[5] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · impliesBefore choose 6 positions out of 7 After C(7,6)=5040/(720·1)=7 Conditions: none Count each k-position subset once, dividing out reorderings inside each of the two groups. Step identifiers and dependenciesRule: subset_count · Uses: G001, S033 S035 · impliesBefore x^1: (-3)^1 After x^1 = -3 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S034 S036 · impliesBefore y^6: (0)^6 After y^6 = 0 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S035 S037 · calculationBefore C·x power (k=6) := mul(7, -3) After C·x power (k=6) = -21 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore term[6] := mul(-21, 0) After term[6] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · impliesBefore choose 7 positions out of 7 After C(7,7)=5040/(5040·1)=1 Conditions: none Count each k-position subset once, dividing out reorderings inside each of the two groups. Step identifiers and dependenciesRule: subset_count · Uses: G001, S038 S040 · impliesBefore x^0: (-3)^0 After x^0 = 1 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S039 S041 · impliesBefore y^7: (0)^7 After y^7 = 0 Conditions: none Evaluate the finite product for a nonnegative integer exponent; the zeroth power uses the multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S040 S042 · calculationBefore C·x power (k=7) := mul(1, 1) After C·x power (k=7) = 1 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore term[7] := mul(1, 0) After term[7] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore expanded sum := sum(-2187, 0, 0, 0, 0, 0, 0, 0) After expanded sum = -2187 Conditions: none Sum every displayed k term, not merely the endpoints. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · impliesBefore direct=-2187; expanded=-2187 After identity holds: Yes Conditions: none The two calculation paths agree on this finite instance of the identity. Step identifiers and dependenciesRule: finite_expansion_evaluation · Uses: G001, S044 ## Reverse checks - S045 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S044 · premise_and_result_audit — Recompute expanded sum with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute term[7] with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute C·x power (k=7) with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S040 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S039 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S038 · premise_and_result_audit — Recompute term[6] with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute C·x power (k=6) with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S035 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S034 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S033 · premise_and_result_audit — Recompute term[5] with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute C·x power (k=5) with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S030 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S029 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S028 · premise_and_result_audit — Recompute term[4] with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute C·x power (k=4) with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S025 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S024 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S023 · premise_and_result_audit — Recompute term[3] with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute C·x power (k=3) with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S020 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S019 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S018 · premise_and_result_audit — Recompute term[2] with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute C·x power (k=2) with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S015 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S014 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S013 · premise_and_result_audit — Recompute term[1] with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute C·x power (k=1) with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S010 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S009 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S008 · premise_and_result_audit — Recompute term[0] with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute C·x power (k=0) with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S005 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S004 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S003 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S002 · premise_and_result_audit — Recompute x+y with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. ## Result (7, -3, 0, -2187, -2187, Yes) The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: e1d6ba36610032034746748fdc8cb00b7f4ba99e5d014cb3d3782cfac6d913b9 - candidate record SHA-256: 6dd1917f1ebf4de68b4f5e4f16015aef136b2f08f5ebee9186c0d417613366ae - semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772 - question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261109-r05-binomial-theorem-0058.json)