Question 448: Compare the Whole Binomial Sum with an Exact Integer Power

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

Step-by-step derivation

n=22≥0; multiplication commutes

group the finite product by the number of y choices

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

x+y = -2

(x+y)^n: (-2)^22

(x+y)^n = 4194304

choose 0 positions out of 22

C(22,0)=1124000727777607680000/(1·1124000727777607680000)=1

x^22: (-2)^22

x^22 = 4194304

y^0: (0)^0

y^0 = 1

C·x power (k=0) := mul(1, 4194304)

C·x power (k=0) = 4194304

term[0] := mul(4194304, 1)

term[0] = 4194304

choose 1 positions out of 22

C(22,1)=1124000727777607680000/(1·51090942171709440000)=22

x^21: (-2)^21

x^21 = -2097152

y^1: (0)^1

y^1 = 0

C·x power (k=1) := mul(22, -2097152)

C·x power (k=1) = -46137344

term[1] := mul(-46137344, 0)

term[1] = 0

choose 2 positions out of 22

C(22,2)=1124000727777607680000/(2·2432902008176640000)=231

x^20: (-2)^20

x^20 = 1048576

y^2: (0)^2

y^2 = 0

C·x power (k=2) := mul(231, 1048576)

C·x power (k=2) = 242221056

term[2] := mul(242221056, 0)

term[2] = 0

choose 3 positions out of 22

C(22,3)=1124000727777607680000/(6·121645100408832000)=1540

x^19: (-2)^19

x^19 = -524288

y^3: (0)^3

y^3 = 0

C·x power (k=3) := mul(1540, -524288)

C·x power (k=3) = -807403520

term[3] := mul(-807403520, 0)

term[3] = 0

choose 4 positions out of 22

C(22,4)=1124000727777607680000/(24·6402373705728000)=7315

x^18: (-2)^18

x^18 = 262144

y^4: (0)^4

y^4 = 0

C·x power (k=4) := mul(7315, 262144)

C·x power (k=4) = 1917583360

term[4] := mul(1917583360, 0)

term[4] = 0

choose 5 positions out of 22

C(22,5)=1124000727777607680000/(120·355687428096000)=26334

x^17: (-2)^17

x^17 = -131072

y^5: (0)^5

y^5 = 0

C·x power (k=5) := mul(26334, -131072)

C·x power (k=5) = -3451650048

term[5] := mul(-3451650048, 0)

term[5] = 0

choose 6 positions out of 22

C(22,6)=1124000727777607680000/(720·20922789888000)=74613

x^16: (-2)^16

x^16 = 65536

y^6: (0)^6

y^6 = 0

C·x power (k=6) := mul(74613, 65536)

C·x power (k=6) = 4889837568

term[6] := mul(4889837568, 0)

term[6] = 0

choose 7 positions out of 22

C(22,7)=1124000727777607680000/(5040·1307674368000)=170544

x^15: (-2)^15

x^15 = -32768

y^7: (0)^7

y^7 = 0

C·x power (k=7) := mul(170544, -32768)

C·x power (k=7) = -5588385792

term[7] := mul(-5588385792, 0)

term[7] = 0

choose 8 positions out of 22

C(22,8)=1124000727777607680000/(40320·87178291200)=319770

x^14: (-2)^14

x^14 = 16384

y^8: (0)^8

y^8 = 0

C·x power (k=8) := mul(319770, 16384)

C·x power (k=8) = 5239111680

term[8] := mul(5239111680, 0)

term[8] = 0

choose 9 positions out of 22

C(22,9)=1124000727777607680000/(362880·6227020800)=497420

x^13: (-2)^13

x^13 = -8192

y^9: (0)^9

y^9 = 0

C·x power (k=9) := mul(497420, -8192)

C·x power (k=9) = -4074864640

term[9] := mul(-4074864640, 0)

term[9] = 0

choose 10 positions out of 22

C(22,10)=1124000727777607680000/(3628800·479001600)=646646

x^12: (-2)^12

x^12 = 4096

y^10: (0)^10

y^10 = 0

C·x power (k=10) := mul(646646, 4096)

C·x power (k=10) = 2648662016

term[10] := mul(2648662016, 0)

term[10] = 0

choose 11 positions out of 22

C(22,11)=1124000727777607680000/(39916800·39916800)=705432

x^11: (-2)^11

x^11 = -2048

y^11: (0)^11

y^11 = 0

C·x power (k=11) := mul(705432, -2048)

C·x power (k=11) = -1444724736

term[11] := mul(-1444724736, 0)

term[11] = 0

choose 12 positions out of 22

C(22,12)=1124000727777607680000/(479001600·3628800)=646646

x^10: (-2)^10

x^10 = 1024

y^12: (0)^12

y^12 = 0

C·x power (k=12) := mul(646646, 1024)

C·x power (k=12) = 662165504

term[12] := mul(662165504, 0)

term[12] = 0

choose 13 positions out of 22

C(22,13)=1124000727777607680000/(6227020800·362880)=497420

x^9: (-2)^9

x^9 = -512

y^13: (0)^13

y^13 = 0

C·x power (k=13) := mul(497420, -512)

C·x power (k=13) = -254679040

term[13] := mul(-254679040, 0)

term[13] = 0

choose 14 positions out of 22

C(22,14)=1124000727777607680000/(87178291200·40320)=319770

x^8: (-2)^8

x^8 = 256

y^14: (0)^14

y^14 = 0

C·x power (k=14) := mul(319770, 256)

C·x power (k=14) = 81861120

term[14] := mul(81861120, 0)

term[14] = 0

choose 15 positions out of 22

C(22,15)=1124000727777607680000/(1307674368000·5040)=170544

x^7: (-2)^7

x^7 = -128

y^15: (0)^15

y^15 = 0

C·x power (k=15) := mul(170544, -128)

C·x power (k=15) = -21829632

term[15] := mul(-21829632, 0)

term[15] = 0

choose 16 positions out of 22

C(22,16)=1124000727777607680000/(20922789888000·720)=74613

x^6: (-2)^6

x^6 = 64

y^16: (0)^16

y^16 = 0

C·x power (k=16) := mul(74613, 64)

C·x power (k=16) = 4775232

term[16] := mul(4775232, 0)

term[16] = 0

choose 17 positions out of 22

C(22,17)=1124000727777607680000/(355687428096000·120)=26334

x^5: (-2)^5

x^5 = -32

y^17: (0)^17

y^17 = 0

C·x power (k=17) := mul(26334, -32)

C·x power (k=17) = -842688

term[17] := mul(-842688, 0)

term[17] = 0

choose 18 positions out of 22

C(22,18)=1124000727777607680000/(6402373705728000·24)=7315

x^4: (-2)^4

x^4 = 16

y^18: (0)^18

y^18 = 0

C·x power (k=18) := mul(7315, 16)

C·x power (k=18) = 117040

term[18] := mul(117040, 0)

term[18] = 0

choose 19 positions out of 22

C(22,19)=1124000727777607680000/(121645100408832000·6)=1540

x^3: (-2)^3

x^3 = -8

y^19: (0)^19

y^19 = 0

C·x power (k=19) := mul(1540, -8)

C·x power (k=19) = -12320

term[19] := mul(-12320, 0)

term[19] = 0

choose 20 positions out of 22

C(22,20)=1124000727777607680000/(2432902008176640000·2)=231

x^2: (-2)^2

x^2 = 4

y^20: (0)^20

y^20 = 0

C·x power (k=20) := mul(231, 4)

C·x power (k=20) = 924

term[20] := mul(924, 0)

term[20] = 0

choose 21 positions out of 22

C(22,21)=1124000727777607680000/(51090942171709440000·1)=22

x^1: (-2)^1

x^1 = -2

y^21: (0)^21

y^21 = 0

C·x power (k=21) := mul(22, -2)

C·x power (k=21) = -44

term[21] := mul(-44, 0)

term[21] = 0

choose 22 positions out of 22

C(22,22)=1124000727777607680000/(1124000727777607680000·1)=1

x^0: (-2)^0

x^0 = 1

y^22: (0)^22

y^22 = 0

C·x power (k=22) := mul(1, 1)

C·x power (k=22) = 1

term[22] := mul(1, 0)

term[22] = 0

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)

expanded sum = 4194304

direct=4194304; expanded=4194304

identity holds: Yes

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.