# Question 439: 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-0064) 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. P1000-0439 · q-20261014-advance-20261109-r05-binomial-theorem-0064 ## Original question and givens Use natural exponent n=13, exact commuting integers x=3, y=-1. Give (n, x, y, power, expansion_sum, equal). - G001 — Use natural exponent n=13, exact commuting integers x=3, y=-1. 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=13≥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, -1) After x+y = 2 Conditions: none First calculate the sum directly. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore (x+y)^n: (2)^13 After (x+y)^n = 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, S002 S004 · impliesBefore choose 0 positions out of 13 After C(13,0)=6227020800/(1·6227020800)=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^13: (3)^13 After x^13 = 1594323 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: (-1)^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, 1594323) After C·x power (k=0) = 1594323 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(1594323, 1) After term[0] = 1594323 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 13 After C(13,1)=6227020800/(1·479001600)=13 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^12: (3)^12 After x^12 = 531441 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: (-1)^1 After y^1 = -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, S010 S012 · calculationBefore C·x power (k=1) := mul(13, 531441) After C·x power (k=1) = 6908733 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(6908733, -1) After term[1] = -6908733 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 13 After C(13,2)=6227020800/(2·39916800)=78 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^11: (3)^11 After x^11 = 177147 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: (-1)^2 After y^2 = 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, S015 S017 · calculationBefore C·x power (k=2) := mul(78, 177147) After C·x power (k=2) = 13817466 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(13817466, 1) After term[2] = 13817466 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 13 After C(13,3)=6227020800/(6·3628800)=286 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^10: (3)^10 After x^10 = 59049 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: (-1)^3 After y^3 = -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, S020 S022 · calculationBefore C·x power (k=3) := mul(286, 59049) After C·x power (k=3) = 16888014 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(16888014, -1) After term[3] = -16888014 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 13 After C(13,4)=6227020800/(24·362880)=715 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^9: (3)^9 After x^9 = 19683 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: (-1)^4 After y^4 = 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, S025 S027 · calculationBefore C·x power (k=4) := mul(715, 19683) After C·x power (k=4) = 14073345 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(14073345, 1) After term[4] = 14073345 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 13 After C(13,5)=6227020800/(120·40320)=1287 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^8: (3)^8 After x^8 = 6561 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: (-1)^5 After y^5 = -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, S030 S032 · calculationBefore C·x power (k=5) := mul(1287, 6561) After C·x power (k=5) = 8444007 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(8444007, -1) After term[5] = -8444007 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 13 After C(13,6)=6227020800/(720·5040)=1716 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^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, S034 S036 · impliesBefore y^6: (-1)^6 After y^6 = 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, S035 S037 · calculationBefore C·x power (k=6) := mul(1716, 2187) After C·x power (k=6) = 3752892 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(3752892, 1) After term[6] = 3752892 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 13 After C(13,7)=6227020800/(5040·720)=1716 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^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, S039 S041 · impliesBefore y^7: (-1)^7 After y^7 = -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, S040 S042 · calculationBefore C·x power (k=7) := mul(1716, 729) After C·x power (k=7) = 1250964 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(1250964, -1) After term[7] = -1250964 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 13 After C(13,8)=6227020800/(40320·120)=1287 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^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, S044 S046 · impliesBefore y^8: (-1)^8 After y^8 = 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, S045 S047 · calculationBefore C·x power (k=8) := mul(1287, 243) After C·x power (k=8) = 312741 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(312741, 1) After term[8] = 312741 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 13 After C(13,9)=6227020800/(362880·24)=715 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^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, S049 S051 · impliesBefore y^9: (-1)^9 After y^9 = -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, S050 S052 · calculationBefore C·x power (k=9) := mul(715, 81) After C·x power (k=9) = 57915 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(57915, -1) After term[9] = -57915 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 13 After C(13,10)=6227020800/(3628800·6)=286 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^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, S054 S056 · impliesBefore y^10: (-1)^10 After y^10 = 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, S055 S057 · calculationBefore C·x power (k=10) := mul(286, 27) After C·x power (k=10) = 7722 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(7722, 1) After term[10] = 7722 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 13 After C(13,11)=6227020800/(39916800·2)=78 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^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, S059 S061 · impliesBefore y^11: (-1)^11 After y^11 = -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, S060 S062 · calculationBefore C·x power (k=11) := mul(78, 9) After C·x power (k=11) = 702 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(702, -1) After term[11] = -702 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 13 After C(13,12)=6227020800/(479001600·1)=13 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^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, S064 S066 · impliesBefore y^12: (-1)^12 After y^12 = 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, S065 S067 · calculationBefore C·x power (k=12) := mul(13, 3) After C·x power (k=12) = 39 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(39, 1) After term[12] = 39 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 13 After C(13,13)=6227020800/(6227020800·1)=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, S068 S070 · 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, S069 S071 · impliesBefore y^13: (-1)^13 After y^13 = -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, S070 S072 · calculationBefore C·x power (k=13) := mul(1, 1) After C·x power (k=13) = 1 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(1, -1) After term[13] = -1 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · calculationBefore expanded sum := sum(1594323, -6908733, 13817466, -16888014, 14073345, -8444007, 3752892, -1250964, 312741, -57915, 7722, -702, 39, -1) After expanded sum = 8192 Conditions: none Sum every displayed k term, not merely the endpoints. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S073 S075 · impliesBefore direct=8192; expanded=8192 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, S074 ## Reverse checks - 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 — Recompute expanded sum with exact rational arithmetic, not rounded decimals. - 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 (13, 3, -1, 8192, 8192, 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: 68709557c411421e722a54bd30726f0d1ffeef1171c77dc814885ac4db5baf81 - candidate record SHA-256: d36c00a03eb2cdf2670afc718ae0b79d3ec2d9ef3f1f6bd81dd830aaac116a71 - 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-0064.json)