# Question 448: 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-0073) 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. P1000-0448 · q-20261014-advance-20261109-r05-binomial-theorem-0073 ## Original question and givens Use natural exponent n=22, exact commuting integers x=-2, y=0. Give (n, x, y, power, expansion_sum, equal). - G001 — Use natural exponent n=22, exact commuting integers x=-2, 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=22≥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, 0) 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)^22 After (x+y)^n = 4194304 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 22 After C(22,0)=1124000727777607680000/(1·1124000727777607680000)=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^22: (-2)^22 After x^22 = 4194304 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, 4194304) After C·x power (k=0) = 4194304 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(4194304, 1) After term[0] = 4194304 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 22 After C(22,1)=1124000727777607680000/(1·51090942171709440000)=22 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^21: (-2)^21 After x^21 = -2097152 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(22, -2097152) After C·x power (k=1) = -46137344 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(-46137344, 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 22 After C(22,2)=1124000727777607680000/(2·2432902008176640000)=231 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^20: (-2)^20 After x^20 = 1048576 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(231, 1048576) After C·x power (k=2) = 242221056 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(242221056, 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 22 After C(22,3)=1124000727777607680000/(6·121645100408832000)=1540 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^19: (-2)^19 After x^19 = -524288 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(1540, -524288) After C·x power (k=3) = -807403520 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(-807403520, 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 22 After C(22,4)=1124000727777607680000/(24·6402373705728000)=7315 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^18: (-2)^18 After x^18 = 262144 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(7315, 262144) After C·x power (k=4) = 1917583360 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(1917583360, 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 22 After C(22,5)=1124000727777607680000/(120·355687428096000)=26334 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^17: (-2)^17 After x^17 = -131072 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(26334, -131072) After C·x power (k=5) = -3451650048 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(-3451650048, 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 22 After C(22,6)=1124000727777607680000/(720·20922789888000)=74613 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^16: (-2)^16 After x^16 = 65536 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(74613, 65536) After C·x power (k=6) = 4889837568 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(4889837568, 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 22 After C(22,7)=1124000727777607680000/(5040·1307674368000)=170544 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^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, 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(170544, -32768) After C·x power (k=7) = -5588385792 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(-5588385792, 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 · impliesBefore choose 8 positions out of 22 After C(22,8)=1124000727777607680000/(40320·87178291200)=319770 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^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, S044 S046 · impliesBefore y^8: (0)^8 After y^8 = 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, S045 S047 · calculationBefore C·x power (k=8) := mul(319770, 16384) After C·x power (k=8) = 5239111680 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(5239111680, 0) After term[8] = 0 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 22 After C(22,9)=1124000727777607680000/(362880·6227020800)=497420 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^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, S049 S051 · impliesBefore y^9: (0)^9 After y^9 = 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, S050 S052 · calculationBefore C·x power (k=9) := mul(497420, -8192) After C·x power (k=9) = -4074864640 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(-4074864640, 0) After term[9] = 0 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 22 After C(22,10)=1124000727777607680000/(3628800·479001600)=646646 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^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, S054 S056 · impliesBefore y^10: (0)^10 After y^10 = 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, S055 S057 · calculationBefore C·x power (k=10) := mul(646646, 4096) After C·x power (k=10) = 2648662016 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(2648662016, 0) After term[10] = 0 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 22 After C(22,11)=1124000727777607680000/(39916800·39916800)=705432 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^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, S059 S061 · impliesBefore y^11: (0)^11 After y^11 = 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, S060 S062 · calculationBefore C·x power (k=11) := mul(705432, -2048) After C·x power (k=11) = -1444724736 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(-1444724736, 0) After term[11] = 0 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 22 After C(22,12)=1124000727777607680000/(479001600·3628800)=646646 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^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, S064 S066 · impliesBefore y^12: (0)^12 After y^12 = 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, S065 S067 · calculationBefore C·x power (k=12) := mul(646646, 1024) After C·x power (k=12) = 662165504 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(662165504, 0) After term[12] = 0 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 22 After C(22,13)=1124000727777607680000/(6227020800·362880)=497420 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^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, S069 S071 · impliesBefore y^13: (0)^13 After y^13 = 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, S070 S072 · calculationBefore C·x power (k=13) := mul(497420, -512) After C·x power (k=13) = -254679040 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(-254679040, 0) After term[13] = 0 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 22 After C(22,14)=1124000727777607680000/(87178291200·40320)=319770 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^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, S074 S076 · impliesBefore y^14: (0)^14 After y^14 = 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, S075 S077 · calculationBefore C·x power (k=14) := mul(319770, 256) After C·x power (k=14) = 81861120 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(81861120, 0) After term[14] = 0 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 22 After C(22,15)=1124000727777607680000/(1307674368000·5040)=170544 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^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, S079 S081 · impliesBefore y^15: (0)^15 After y^15 = 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, S080 S082 · calculationBefore C·x power (k=15) := mul(170544, -128) After C·x power (k=15) = -21829632 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(-21829632, 0) After term[15] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · impliesBefore choose 16 positions out of 22 After C(22,16)=1124000727777607680000/(20922789888000·720)=74613 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, S083 S085 · 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, S084 S086 · impliesBefore y^16: (0)^16 After y^16 = 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, S085 S087 · calculationBefore C·x power (k=16) := mul(74613, 64) After C·x power (k=16) = 4775232 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S086 S088 · calculationBefore term[16] := mul(4775232, 0) After term[16] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S087 S089 · impliesBefore choose 17 positions out of 22 After C(22,17)=1124000727777607680000/(355687428096000·120)=26334 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, S088 S090 · 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, S089 S091 · impliesBefore y^17: (0)^17 After y^17 = 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, S090 S092 · calculationBefore C·x power (k=17) := mul(26334, -32) After C·x power (k=17) = -842688 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S091 S093 · calculationBefore term[17] := mul(-842688, 0) After term[17] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S092 S094 · impliesBefore choose 18 positions out of 22 After C(22,18)=1124000727777607680000/(6402373705728000·24)=7315 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, S093 S095 · 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, S094 S096 · impliesBefore y^18: (0)^18 After y^18 = 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, S095 S097 · calculationBefore C·x power (k=18) := mul(7315, 16) After C·x power (k=18) = 117040 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S096 S098 · calculationBefore term[18] := mul(117040, 0) After term[18] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S097 S099 · impliesBefore choose 19 positions out of 22 After C(22,19)=1124000727777607680000/(121645100408832000·6)=1540 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, S098 S100 · 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, S099 S101 · impliesBefore y^19: (0)^19 After y^19 = 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, S100 S102 · calculationBefore C·x power (k=19) := mul(1540, -8) After C·x power (k=19) = -12320 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S101 S103 · calculationBefore term[19] := mul(-12320, 0) After term[19] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S102 S104 · impliesBefore choose 20 positions out of 22 After C(22,20)=1124000727777607680000/(2432902008176640000·2)=231 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, S103 S105 · 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, S104 S106 · impliesBefore y^20: (0)^20 After y^20 = 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, S105 S107 · calculationBefore C·x power (k=20) := mul(231, 4) After C·x power (k=20) = 924 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S106 S108 · calculationBefore term[20] := mul(924, 0) After term[20] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S107 S109 · impliesBefore choose 21 positions out of 22 After C(22,21)=1124000727777607680000/(51090942171709440000·1)=22 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, S108 S110 · 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, S109 S111 · impliesBefore y^21: (0)^21 After y^21 = 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, S110 S112 · calculationBefore C·x power (k=21) := mul(22, -2) After C·x power (k=21) = -44 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S111 S113 · calculationBefore term[21] := mul(-44, 0) After term[21] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S112 S114 · impliesBefore choose 22 positions out of 22 After C(22,22)=1124000727777607680000/(1124000727777607680000·1)=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, S113 S115 · 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, S114 S116 · impliesBefore y^22: (0)^22 After y^22 = 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, S115 S117 · calculationBefore C·x power (k=22) := mul(1, 1) After C·x power (k=22) = 1 Conditions: none Multiply the coefficient by the x power first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S116 S118 · calculationBefore term[22] := mul(1, 0) After term[22] = 0 Conditions: none Multiply by the y power, preserving parentheses around a negative base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S117 S119 · calculationBefore expanded sum := sum(4194304, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) After expanded sum = 4194304 Conditions: none Sum every displayed k term, not merely the endpoints. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S118 S120 · impliesBefore direct=4194304; expanded=4194304 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, S119 ## Reverse checks - S120 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S119 · premise_and_result_audit — Recompute expanded sum with exact rational arithmetic, not rounded decimals. - S118 · premise_and_result_audit — Recompute term[22] with exact rational arithmetic, not rounded decimals. - S117 · premise_and_result_audit — Recompute C·x power (k=22) with exact rational arithmetic, not rounded decimals. - S116 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S115 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S114 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S113 · premise_and_result_audit — Recompute term[21] with exact rational arithmetic, not rounded decimals. - S112 · premise_and_result_audit — Recompute C·x power (k=21) with exact rational arithmetic, not rounded decimals. - S111 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S110 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S109 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S108 · premise_and_result_audit — Recompute term[20] with exact rational arithmetic, not rounded decimals. - S107 · premise_and_result_audit — Recompute C·x power (k=20) with exact rational arithmetic, not rounded decimals. - S106 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S105 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S104 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S103 · premise_and_result_audit — Recompute term[19] with exact rational arithmetic, not rounded decimals. - S102 · premise_and_result_audit — Recompute C·x power (k=19) with exact rational arithmetic, not rounded decimals. - S101 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S100 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S099 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S098 · premise_and_result_audit — Recompute term[18] with exact rational arithmetic, not rounded decimals. - S097 · premise_and_result_audit — Recompute C·x power (k=18) with exact rational arithmetic, not rounded decimals. - S096 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S095 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S094 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S093 · premise_and_result_audit — Recompute term[17] with exact rational arithmetic, not rounded decimals. - S092 · premise_and_result_audit — Recompute C·x power (k=17) with exact rational arithmetic, not rounded decimals. - S091 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S090 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S089 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - S088 · premise_and_result_audit — Recompute term[16] with exact rational arithmetic, not rounded decimals. - S087 · premise_and_result_audit — Recompute C·x power (k=16) with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - 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 — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring. - 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 (22, -2, 0, 4194304, 4194304, 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: 1a43d8fd7a7301f1cc5c2b7c50d7baeef29e7b4e7d9d7e2018619fb9bb88e286 - candidate record SHA-256: 9bd62545dc014f7778d590b977d8650df13f76013e443f5adc88fb149720c814 - 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-0073.json)