Question 450: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=24, exact commuting integers x=0, y=-2. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=24, exact commuting integers x=0, 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

Step-by-step derivation

n=24≥0; multiplication commutes

group the finite product by the number of y choices

x+y := sum(0, -2)

x+y = -2

(x+y)^n: (-2)^24

(x+y)^n = 16777216

choose 0 positions out of 24

C(24,0)=620448401733239439360000/(1·620448401733239439360000)=1

x^24: (0)^24

x^24 = 0

y^0: (-2)^0

y^0 = 1

C·x power (k=0) := mul(1, 0)

C·x power (k=0) = 0

term[0] := mul(0, 1)

term[0] = 0

choose 1 positions out of 24

C(24,1)=620448401733239439360000/(1·25852016738884976640000)=24

x^23: (0)^23

x^23 = 0

y^1: (-2)^1

y^1 = -2

C·x power (k=1) := mul(24, 0)

C·x power (k=1) = 0

term[1] := mul(0, -2)

term[1] = 0

choose 2 positions out of 24

C(24,2)=620448401733239439360000/(2·1124000727777607680000)=276

x^22: (0)^22

x^22 = 0

y^2: (-2)^2

y^2 = 4

C·x power (k=2) := mul(276, 0)

C·x power (k=2) = 0

term[2] := mul(0, 4)

term[2] = 0

choose 3 positions out of 24

C(24,3)=620448401733239439360000/(6·51090942171709440000)=2024

x^21: (0)^21

x^21 = 0

y^3: (-2)^3

y^3 = -8

C·x power (k=3) := mul(2024, 0)

C·x power (k=3) = 0

term[3] := mul(0, -8)

term[3] = 0

choose 4 positions out of 24

C(24,4)=620448401733239439360000/(24·2432902008176640000)=10626

x^20: (0)^20

x^20 = 0

y^4: (-2)^4

y^4 = 16

C·x power (k=4) := mul(10626, 0)

C·x power (k=4) = 0

term[4] := mul(0, 16)

term[4] = 0

choose 5 positions out of 24

C(24,5)=620448401733239439360000/(120·121645100408832000)=42504

x^19: (0)^19

x^19 = 0

y^5: (-2)^5

y^5 = -32

C·x power (k=5) := mul(42504, 0)

C·x power (k=5) = 0

term[5] := mul(0, -32)

term[5] = 0

choose 6 positions out of 24

C(24,6)=620448401733239439360000/(720·6402373705728000)=134596

x^18: (0)^18

x^18 = 0

y^6: (-2)^6

y^6 = 64

C·x power (k=6) := mul(134596, 0)

C·x power (k=6) = 0

term[6] := mul(0, 64)

term[6] = 0

choose 7 positions out of 24

C(24,7)=620448401733239439360000/(5040·355687428096000)=346104

x^17: (0)^17

x^17 = 0

y^7: (-2)^7

y^7 = -128

C·x power (k=7) := mul(346104, 0)

C·x power (k=7) = 0

term[7] := mul(0, -128)

term[7] = 0

choose 8 positions out of 24

C(24,8)=620448401733239439360000/(40320·20922789888000)=735471

x^16: (0)^16

x^16 = 0

y^8: (-2)^8

y^8 = 256

C·x power (k=8) := mul(735471, 0)

C·x power (k=8) = 0

term[8] := mul(0, 256)

term[8] = 0

choose 9 positions out of 24

C(24,9)=620448401733239439360000/(362880·1307674368000)=1307504

x^15: (0)^15

x^15 = 0

y^9: (-2)^9

y^9 = -512

C·x power (k=9) := mul(1307504, 0)

C·x power (k=9) = 0

term[9] := mul(0, -512)

term[9] = 0

choose 10 positions out of 24

C(24,10)=620448401733239439360000/(3628800·87178291200)=1961256

x^14: (0)^14

x^14 = 0

y^10: (-2)^10

y^10 = 1024

C·x power (k=10) := mul(1961256, 0)

C·x power (k=10) = 0

term[10] := mul(0, 1024)

term[10] = 0

choose 11 positions out of 24

C(24,11)=620448401733239439360000/(39916800·6227020800)=2496144

x^13: (0)^13

x^13 = 0

y^11: (-2)^11

y^11 = -2048

C·x power (k=11) := mul(2496144, 0)

C·x power (k=11) = 0

term[11] := mul(0, -2048)

term[11] = 0

choose 12 positions out of 24

C(24,12)=620448401733239439360000/(479001600·479001600)=2704156

x^12: (0)^12

x^12 = 0

y^12: (-2)^12

y^12 = 4096

C·x power (k=12) := mul(2704156, 0)

C·x power (k=12) = 0

term[12] := mul(0, 4096)

term[12] = 0

choose 13 positions out of 24

C(24,13)=620448401733239439360000/(6227020800·39916800)=2496144

x^11: (0)^11

x^11 = 0

y^13: (-2)^13

y^13 = -8192

C·x power (k=13) := mul(2496144, 0)

C·x power (k=13) = 0

term[13] := mul(0, -8192)

term[13] = 0

choose 14 positions out of 24

C(24,14)=620448401733239439360000/(87178291200·3628800)=1961256

x^10: (0)^10

x^10 = 0

y^14: (-2)^14

y^14 = 16384

C·x power (k=14) := mul(1961256, 0)

C·x power (k=14) = 0

term[14] := mul(0, 16384)

term[14] = 0

choose 15 positions out of 24

C(24,15)=620448401733239439360000/(1307674368000·362880)=1307504

x^9: (0)^9

x^9 = 0

y^15: (-2)^15

y^15 = -32768

C·x power (k=15) := mul(1307504, 0)

C·x power (k=15) = 0

term[15] := mul(0, -32768)

term[15] = 0

choose 16 positions out of 24

C(24,16)=620448401733239439360000/(20922789888000·40320)=735471

x^8: (0)^8

x^8 = 0

y^16: (-2)^16

y^16 = 65536

C·x power (k=16) := mul(735471, 0)

C·x power (k=16) = 0

term[16] := mul(0, 65536)

term[16] = 0

choose 17 positions out of 24

C(24,17)=620448401733239439360000/(355687428096000·5040)=346104

x^7: (0)^7

x^7 = 0

y^17: (-2)^17

y^17 = -131072

C·x power (k=17) := mul(346104, 0)

C·x power (k=17) = 0

term[17] := mul(0, -131072)

term[17] = 0

choose 18 positions out of 24

C(24,18)=620448401733239439360000/(6402373705728000·720)=134596

x^6: (0)^6

x^6 = 0

y^18: (-2)^18

y^18 = 262144

C·x power (k=18) := mul(134596, 0)

C·x power (k=18) = 0

term[18] := mul(0, 262144)

term[18] = 0

choose 19 positions out of 24

C(24,19)=620448401733239439360000/(121645100408832000·120)=42504

x^5: (0)^5

x^5 = 0

y^19: (-2)^19

y^19 = -524288

C·x power (k=19) := mul(42504, 0)

C·x power (k=19) = 0

term[19] := mul(0, -524288)

term[19] = 0

choose 20 positions out of 24

C(24,20)=620448401733239439360000/(2432902008176640000·24)=10626

x^4: (0)^4

x^4 = 0

y^20: (-2)^20

y^20 = 1048576

C·x power (k=20) := mul(10626, 0)

C·x power (k=20) = 0

term[20] := mul(0, 1048576)

term[20] = 0

choose 21 positions out of 24

C(24,21)=620448401733239439360000/(51090942171709440000·6)=2024

x^3: (0)^3

x^3 = 0

y^21: (-2)^21

y^21 = -2097152

C·x power (k=21) := mul(2024, 0)

C·x power (k=21) = 0

term[21] := mul(0, -2097152)

term[21] = 0

choose 22 positions out of 24

C(24,22)=620448401733239439360000/(1124000727777607680000·2)=276

x^2: (0)^2

x^2 = 0

y^22: (-2)^22

y^22 = 4194304

C·x power (k=22) := mul(276, 0)

C·x power (k=22) = 0

term[22] := mul(0, 4194304)

term[22] = 0

choose 23 positions out of 24

C(24,23)=620448401733239439360000/(25852016738884976640000·1)=24

x^1: (0)^1

x^1 = 0

y^23: (-2)^23

y^23 = -8388608

C·x power (k=23) := mul(24, 0)

C·x power (k=23) = 0

term[23] := mul(0, -8388608)

term[23] = 0

choose 24 positions out of 24

C(24,24)=620448401733239439360000/(620448401733239439360000·1)=1

x^0: (0)^0

x^0 = 1

y^24: (-2)^24

y^24 = 16777216

C·x power (k=24) := mul(1, 1)

C·x power (k=24) = 1

term[24] := mul(1, 16777216)

term[24] = 16777216

expanded sum := sum(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16777216)

expanded sum = 16777216

direct=16777216; expanded=16777216

identity holds: Yes

Reverse checks

• S130 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S129 · premise_and_result_audit — Recompute expanded sum with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute term[24] with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute C·x power (k=24) with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S125 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S124 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S123 · premise_and_result_audit — Recompute term[23] with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Recompute C·x power (k=23) with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• 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 — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• 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

(24, 0, -2, 16777216, 16777216, 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.