# Question 441: 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-0066) 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. P1000-0441 · q-20261014-advance-20261109-r05-binomial-theorem-0066 ## Original question and givens Use natural exponent n=15, exact commuting integers x=-2, y=2. Give (n, x, y, power, expansion_sum, equal). - G001 — Use natural exponent n=15, exact commuting integers x=-2, y=2. 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=15≥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(-2, 2) After x+y = 0 Conditions: none First calculate the sum directly. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore (x+y)^n: (0)^15 After (x+y)^n = 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, S002 S004 · impliesBefore choose 0 positions out of 15 After C(15,0)=1307674368000/(1·1307674368000)=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^15: (-2)^15 After x^15 = -32768 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: (2)^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, -32768) After C·x power (k=0) = -32768 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(-32768, 1) After term[0] = -32768 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 15 After C(15,1)=1307674368000/(1·87178291200)=15 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^14: (-2)^14 After x^14 = 16384 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: (2)^1 After y^1 = 2 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(15, 16384) After C·x power (k=1) = 245760 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(245760, 2) After term[1] = 491520 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 15 After C(15,2)=1307674368000/(2·6227020800)=105 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^13: (-2)^13 After x^13 = -8192 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: (2)^2 After y^2 = 4 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(105, -8192) After C·x power (k=2) = -860160 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(-860160, 4) After term[2] = -3440640 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 15 After C(15,3)=1307674368000/(6·479001600)=455 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^12: (-2)^12 After x^12 = 4096 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: (2)^3 After y^3 = 8 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(455, 4096) After C·x power (k=3) = 1863680 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(1863680, 8) After term[3] = 14909440 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 15 After C(15,4)=1307674368000/(24·39916800)=1365 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^11: (-2)^11 After x^11 = -2048 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: (2)^4 After y^4 = 16 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(1365, -2048) After C·x power (k=4) = -2795520 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(-2795520, 16) After term[4] = -44728320 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 15 After C(15,5)=1307674368000/(120·3628800)=3003 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^10: (-2)^10 After x^10 = 1024 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: (2)^5 After y^5 = 32 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(3003, 1024) After C·x power (k=5) = 3075072 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(3075072, 32) After term[5] = 98402304 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 15 After C(15,6)=1307674368000/(720·362880)=5005 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^9: (-2)^9 After x^9 = -512 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: (2)^6 After y^6 = 64 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(5005, -512) After C·x power (k=6) = -2562560 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(-2562560, 64) After term[6] = -164003840 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 15 After C(15,7)=1307674368000/(5040·40320)=6435 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^8: (-2)^8 After x^8 = 256 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: (2)^7 After y^7 = 128 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(6435, 256) After C·x power (k=7) = 1647360 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(1647360, 128) After term[7] = 210862080 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · impliesBefore choose 8 positions out of 15 After C(15,8)=1307674368000/(40320·5040)=6435 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, S043 S045 · impliesBefore x^7: (-2)^7 After x^7 = -128 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, S044 S046 · impliesBefore y^8: (2)^8 After y^8 = 256 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, S045 S047 · calculationBefore C·x power (k=8) := mul(6435, -128) After C·x power (k=8) = -823680 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore term[8] := mul(-823680, 256) After term[8] = -210862080 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · impliesBefore choose 9 positions out of 15 After C(15,9)=1307674368000/(362880·720)=5005 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, S048 S050 · impliesBefore x^6: (-2)^6 After x^6 = 64 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, S049 S051 · impliesBefore y^9: (2)^9 After y^9 = 512 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, S050 S052 · calculationBefore C·x power (k=9) := mul(5005, 64) After C·x power (k=9) = 320320 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore term[9] := mul(320320, 512) After term[9] = 164003840 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · impliesBefore choose 10 positions out of 15 After C(15,10)=1307674368000/(3628800·120)=3003 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, S053 S055 · impliesBefore x^5: (-2)^5 After x^5 = -32 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, S054 S056 · impliesBefore y^10: (2)^10 After y^10 = 1024 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, S055 S057 · calculationBefore C·x power (k=10) := mul(3003, -32) After C·x power (k=10) = -96096 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · calculationBefore term[10] := mul(-96096, 1024) After term[10] = -98402304 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · impliesBefore choose 11 positions out of 15 After C(15,11)=1307674368000/(39916800·24)=1365 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, S058 S060 · impliesBefore x^4: (-2)^4 After x^4 = 16 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, S059 S061 · impliesBefore y^11: (2)^11 After y^11 = 2048 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, S060 S062 · calculationBefore C·x power (k=11) := mul(1365, 16) After C·x power (k=11) = 21840 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · calculationBefore term[11] := mul(21840, 2048) After term[11] = 44728320 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062 S064 · impliesBefore choose 12 positions out of 15 After C(15,12)=1307674368000/(479001600·6)=455 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, S063 S065 · impliesBefore x^3: (-2)^3 After x^3 = -8 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, S064 S066 · impliesBefore y^12: (2)^12 After y^12 = 4096 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, S065 S067 · calculationBefore C·x power (k=12) := mul(455, -8) After C·x power (k=12) = -3640 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore term[12] := mul(-3640, 4096) After term[12] = -14909440 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · impliesBefore choose 13 positions out of 15 After C(15,13)=1307674368000/(6227020800·2)=105 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, S068 S070 · impliesBefore x^2: (-2)^2 After x^2 = 4 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, S069 S071 · impliesBefore y^13: (2)^13 After y^13 = 8192 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, S070 S072 · calculationBefore C·x power (k=13) := mul(105, 4) After C·x power (k=13) = 420 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore term[13] := mul(420, 8192) After term[13] = 3440640 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · impliesBefore choose 14 positions out of 15 After C(15,14)=1307674368000/(87178291200·1)=15 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, S073 S075 · impliesBefore x^1: (-2)^1 After x^1 = -2 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, S074 S076 · impliesBefore y^14: (2)^14 After y^14 = 16384 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, S075 S077 · calculationBefore C·x power (k=14) := mul(15, -2) After C·x power (k=14) = -30 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · calculationBefore term[14] := mul(-30, 16384) After term[14] = -491520 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S077 S079 · impliesBefore choose 15 positions out of 15 After C(15,15)=1307674368000/(1307674368000·1)=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, S078 S080 · impliesBefore x^0: (-2)^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, S079 S081 · impliesBefore y^15: (2)^15 After y^15 = 32768 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, S080 S082 · calculationBefore C·x power (k=15) := mul(1, 1) After C·x power (k=15) = 1 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S081 S083 · calculationBefore term[15] := mul(1, 32768) After term[15] = 32768 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · calculationBefore expanded sum := sum(-32768, 491520, -3440640, 14909440, -44728320, 98402304, -164003840, 210862080, -210862080, 164003840, -98402304, 44728320, -14909440, 3440640, -491520, 32768) After expanded sum = 0 Conditions: none Sum every displayed k term, not merely the endpoints. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083 S085 · impliesBefore direct=0; expanded=0 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, S084 ## Reverse checks - S085 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S084 · premise_and_result_audit — Recompute expanded sum with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute term[15] with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute C·x power (k=15) with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S080 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S079 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S078 · premise_and_result_audit — Recompute term[14] with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Recompute C·x power (k=14) with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S075 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S074 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S073 · premise_and_result_audit — Recompute term[13] with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute C·x power (k=13) with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S070 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S069 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S068 · premise_and_result_audit — Recompute term[12] with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute C·x power (k=12) with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S065 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S064 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S063 · premise_and_result_audit — Recompute term[11] with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute C·x power (k=11) with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S060 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S059 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S058 · premise_and_result_audit — Recompute term[10] with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute C·x power (k=10) with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S055 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S054 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S053 · premise_and_result_audit — Recompute term[9] with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute C·x power (k=9) with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S050 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S049 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S048 · premise_and_result_audit — Recompute term[8] with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute C·x power (k=8) with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - 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 — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - 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 (15, -2, 2, 0, 0, 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: 0bf27582977bcc4a12ddbfe9d88e761188c39892d606e39a0d4bc0eb2d1a0180 - candidate record SHA-256: 45bf51d2479a95d624884cf1bd87d415d8446b1aa3df466a73226c4a0989a896 - 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-0066.json)