Question 445: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=19, exact commuting integers x=2, y=-2. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=19, exact commuting integers x=2, y=-2. Give (n, x, y, power, expansion_sum, equal).

Definition and scope

n is a nonnegative integer and x,y commute. Sum every term C(n,k)x^(n−k)y^k; the zeroth power is 1.

• mko-binomial-theorem

Step-by-step derivation

n=19≥0; multiplication commutes

group the finite product by the number of y choices

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

x+y = 0

(x+y)^n: (0)^19

(x+y)^n = 0

choose 0 positions out of 19

C(19,0)=121645100408832000/(1·121645100408832000)=1

x^19: (2)^19

x^19 = 524288

y^0: (-2)^0

y^0 = 1

C·x power (k=0) := mul(1, 524288)

C·x power (k=0) = 524288

term[0] := mul(524288, 1)

term[0] = 524288

choose 1 positions out of 19

C(19,1)=121645100408832000/(1·6402373705728000)=19

x^18: (2)^18

x^18 = 262144

y^1: (-2)^1

y^1 = -2

C·x power (k=1) := mul(19, 262144)

C·x power (k=1) = 4980736

term[1] := mul(4980736, -2)

term[1] = -9961472

choose 2 positions out of 19

C(19,2)=121645100408832000/(2·355687428096000)=171

x^17: (2)^17

x^17 = 131072

y^2: (-2)^2

y^2 = 4

C·x power (k=2) := mul(171, 131072)

C·x power (k=2) = 22413312

term[2] := mul(22413312, 4)

term[2] = 89653248

choose 3 positions out of 19

C(19,3)=121645100408832000/(6·20922789888000)=969

x^16: (2)^16

x^16 = 65536

y^3: (-2)^3

y^3 = -8

C·x power (k=3) := mul(969, 65536)

C·x power (k=3) = 63504384

term[3] := mul(63504384, -8)

term[3] = -508035072

choose 4 positions out of 19

C(19,4)=121645100408832000/(24·1307674368000)=3876

x^15: (2)^15

x^15 = 32768

y^4: (-2)^4

y^4 = 16

C·x power (k=4) := mul(3876, 32768)

C·x power (k=4) = 127008768

term[4] := mul(127008768, 16)

term[4] = 2032140288

choose 5 positions out of 19

C(19,5)=121645100408832000/(120·87178291200)=11628

x^14: (2)^14

x^14 = 16384

y^5: (-2)^5

y^5 = -32

C·x power (k=5) := mul(11628, 16384)

C·x power (k=5) = 190513152

term[5] := mul(190513152, -32)

term[5] = -6096420864

choose 6 positions out of 19

C(19,6)=121645100408832000/(720·6227020800)=27132

x^13: (2)^13

x^13 = 8192

y^6: (-2)^6

y^6 = 64

C·x power (k=6) := mul(27132, 8192)

C·x power (k=6) = 222265344

term[6] := mul(222265344, 64)

term[6] = 14224982016

choose 7 positions out of 19

C(19,7)=121645100408832000/(5040·479001600)=50388

x^12: (2)^12

x^12 = 4096

y^7: (-2)^7

y^7 = -128

C·x power (k=7) := mul(50388, 4096)

C·x power (k=7) = 206389248

term[7] := mul(206389248, -128)

term[7] = -26417823744

choose 8 positions out of 19

C(19,8)=121645100408832000/(40320·39916800)=75582

x^11: (2)^11

x^11 = 2048

y^8: (-2)^8

y^8 = 256

C·x power (k=8) := mul(75582, 2048)

C·x power (k=8) = 154791936

term[8] := mul(154791936, 256)

term[8] = 39626735616

choose 9 positions out of 19

C(19,9)=121645100408832000/(362880·3628800)=92378

x^10: (2)^10

x^10 = 1024

y^9: (-2)^9

y^9 = -512

C·x power (k=9) := mul(92378, 1024)

C·x power (k=9) = 94595072

term[9] := mul(94595072, -512)

term[9] = -48432676864

choose 10 positions out of 19

C(19,10)=121645100408832000/(3628800·362880)=92378

x^9: (2)^9

x^9 = 512

y^10: (-2)^10

y^10 = 1024

C·x power (k=10) := mul(92378, 512)

C·x power (k=10) = 47297536

term[10] := mul(47297536, 1024)

term[10] = 48432676864

choose 11 positions out of 19

C(19,11)=121645100408832000/(39916800·40320)=75582

x^8: (2)^8

x^8 = 256

y^11: (-2)^11

y^11 = -2048

C·x power (k=11) := mul(75582, 256)

C·x power (k=11) = 19348992

term[11] := mul(19348992, -2048)

term[11] = -39626735616

choose 12 positions out of 19

C(19,12)=121645100408832000/(479001600·5040)=50388

x^7: (2)^7

x^7 = 128

y^12: (-2)^12

y^12 = 4096

C·x power (k=12) := mul(50388, 128)

C·x power (k=12) = 6449664

term[12] := mul(6449664, 4096)

term[12] = 26417823744

choose 13 positions out of 19

C(19,13)=121645100408832000/(6227020800·720)=27132

x^6: (2)^6

x^6 = 64

y^13: (-2)^13

y^13 = -8192

C·x power (k=13) := mul(27132, 64)

C·x power (k=13) = 1736448

term[13] := mul(1736448, -8192)

term[13] = -14224982016

choose 14 positions out of 19

C(19,14)=121645100408832000/(87178291200·120)=11628

x^5: (2)^5

x^5 = 32

y^14: (-2)^14

y^14 = 16384

C·x power (k=14) := mul(11628, 32)

C·x power (k=14) = 372096

term[14] := mul(372096, 16384)

term[14] = 6096420864

choose 15 positions out of 19

C(19,15)=121645100408832000/(1307674368000·24)=3876

x^4: (2)^4

x^4 = 16

y^15: (-2)^15

y^15 = -32768

C·x power (k=15) := mul(3876, 16)

C·x power (k=15) = 62016

term[15] := mul(62016, -32768)

term[15] = -2032140288

choose 16 positions out of 19

C(19,16)=121645100408832000/(20922789888000·6)=969

x^3: (2)^3

x^3 = 8

y^16: (-2)^16

y^16 = 65536

C·x power (k=16) := mul(969, 8)

C·x power (k=16) = 7752

term[16] := mul(7752, 65536)

term[16] = 508035072

choose 17 positions out of 19

C(19,17)=121645100408832000/(355687428096000·2)=171

x^2: (2)^2

x^2 = 4

y^17: (-2)^17

y^17 = -131072

C·x power (k=17) := mul(171, 4)

C·x power (k=17) = 684

term[17] := mul(684, -131072)

term[17] = -89653248

choose 18 positions out of 19

C(19,18)=121645100408832000/(6402373705728000·1)=19

x^1: (2)^1

x^1 = 2

y^18: (-2)^18

y^18 = 262144

C·x power (k=18) := mul(19, 2)

C·x power (k=18) = 38

term[18] := mul(38, 262144)

term[18] = 9961472

choose 19 positions out of 19

C(19,19)=121645100408832000/(121645100408832000·1)=1

x^0: (2)^0

x^0 = 1

y^19: (-2)^19

y^19 = -524288

C·x power (k=19) := mul(1, 1)

C·x power (k=19) = 1

term[19] := mul(1, -524288)

term[19] = -524288

expanded sum := sum(524288, -9961472, 89653248, -508035072, 2032140288, -6096420864, 14224982016, -26417823744, 39626735616, -48432676864, 48432676864, -39626735616, 26417823744, -14224982016, 6096420864, -2032140288, 508035072, -89653248, 9961472, -524288)

expanded sum = 0

direct=0; expanded=0

identity holds: Yes

Reverse checks

• 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 — Recompute expanded sum with exact rational arithmetic, not rounded decimals.
• 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

(19, 2, -2, 0, 0, Yes)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.