Question 449: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=23, exact commuting integers x=-1, y=-1. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=23, exact commuting integers x=-1, y=-1. Give (n, x, y, power, expansion_sum, equal).

Definition and scope

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

• mko-binomial-theorem

Step-by-step derivation

n=23≥0; multiplication commutes

group the finite product by the number of y choices

x+y := sum(-1, -1)

x+y = -2

(x+y)^n: (-2)^23

(x+y)^n = -8388608

choose 0 positions out of 23

C(23,0)=25852016738884976640000/(1·25852016738884976640000)=1

x^23: (-1)^23

x^23 = -1

y^0: (-1)^0

y^0 = 1

C·x power (k=0) := mul(1, -1)

C·x power (k=0) = -1

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

term[0] = -1

choose 1 positions out of 23

C(23,1)=25852016738884976640000/(1·1124000727777607680000)=23

x^22: (-1)^22

x^22 = 1

y^1: (-1)^1

y^1 = -1

C·x power (k=1) := mul(23, 1)

C·x power (k=1) = 23

term[1] := mul(23, -1)

term[1] = -23

choose 2 positions out of 23

C(23,2)=25852016738884976640000/(2·51090942171709440000)=253

x^21: (-1)^21

x^21 = -1

y^2: (-1)^2

y^2 = 1

C·x power (k=2) := mul(253, -1)

C·x power (k=2) = -253

term[2] := mul(-253, 1)

term[2] = -253

choose 3 positions out of 23

C(23,3)=25852016738884976640000/(6·2432902008176640000)=1771

x^20: (-1)^20

x^20 = 1

y^3: (-1)^3

y^3 = -1

C·x power (k=3) := mul(1771, 1)

C·x power (k=3) = 1771

term[3] := mul(1771, -1)

term[3] = -1771

choose 4 positions out of 23

C(23,4)=25852016738884976640000/(24·121645100408832000)=8855

x^19: (-1)^19

x^19 = -1

y^4: (-1)^4

y^4 = 1

C·x power (k=4) := mul(8855, -1)

C·x power (k=4) = -8855

term[4] := mul(-8855, 1)

term[4] = -8855

choose 5 positions out of 23

C(23,5)=25852016738884976640000/(120·6402373705728000)=33649

x^18: (-1)^18

x^18 = 1

y^5: (-1)^5

y^5 = -1

C·x power (k=5) := mul(33649, 1)

C·x power (k=5) = 33649

term[5] := mul(33649, -1)

term[5] = -33649

choose 6 positions out of 23

C(23,6)=25852016738884976640000/(720·355687428096000)=100947

x^17: (-1)^17

x^17 = -1

y^6: (-1)^6

y^6 = 1

C·x power (k=6) := mul(100947, -1)

C·x power (k=6) = -100947

term[6] := mul(-100947, 1)

term[6] = -100947

choose 7 positions out of 23

C(23,7)=25852016738884976640000/(5040·20922789888000)=245157

x^16: (-1)^16

x^16 = 1

y^7: (-1)^7

y^7 = -1

C·x power (k=7) := mul(245157, 1)

C·x power (k=7) = 245157

term[7] := mul(245157, -1)

term[7] = -245157

choose 8 positions out of 23

C(23,8)=25852016738884976640000/(40320·1307674368000)=490314

x^15: (-1)^15

x^15 = -1

y^8: (-1)^8

y^8 = 1

C·x power (k=8) := mul(490314, -1)

C·x power (k=8) = -490314

term[8] := mul(-490314, 1)

term[8] = -490314

choose 9 positions out of 23

C(23,9)=25852016738884976640000/(362880·87178291200)=817190

x^14: (-1)^14

x^14 = 1

y^9: (-1)^9

y^9 = -1

C·x power (k=9) := mul(817190, 1)

C·x power (k=9) = 817190

term[9] := mul(817190, -1)

term[9] = -817190

choose 10 positions out of 23

C(23,10)=25852016738884976640000/(3628800·6227020800)=1144066

x^13: (-1)^13

x^13 = -1

y^10: (-1)^10

y^10 = 1

C·x power (k=10) := mul(1144066, -1)

C·x power (k=10) = -1144066

term[10] := mul(-1144066, 1)

term[10] = -1144066

choose 11 positions out of 23

C(23,11)=25852016738884976640000/(39916800·479001600)=1352078

x^12: (-1)^12

x^12 = 1

y^11: (-1)^11

y^11 = -1

C·x power (k=11) := mul(1352078, 1)

C·x power (k=11) = 1352078

term[11] := mul(1352078, -1)

term[11] = -1352078

choose 12 positions out of 23

C(23,12)=25852016738884976640000/(479001600·39916800)=1352078

x^11: (-1)^11

x^11 = -1

y^12: (-1)^12

y^12 = 1

C·x power (k=12) := mul(1352078, -1)

C·x power (k=12) = -1352078

term[12] := mul(-1352078, 1)

term[12] = -1352078

choose 13 positions out of 23

C(23,13)=25852016738884976640000/(6227020800·3628800)=1144066

x^10: (-1)^10

x^10 = 1

y^13: (-1)^13

y^13 = -1

C·x power (k=13) := mul(1144066, 1)

C·x power (k=13) = 1144066

term[13] := mul(1144066, -1)

term[13] = -1144066

choose 14 positions out of 23

C(23,14)=25852016738884976640000/(87178291200·362880)=817190

x^9: (-1)^9

x^9 = -1

y^14: (-1)^14

y^14 = 1

C·x power (k=14) := mul(817190, -1)

C·x power (k=14) = -817190

term[14] := mul(-817190, 1)

term[14] = -817190

choose 15 positions out of 23

C(23,15)=25852016738884976640000/(1307674368000·40320)=490314

x^8: (-1)^8

x^8 = 1

y^15: (-1)^15

y^15 = -1

C·x power (k=15) := mul(490314, 1)

C·x power (k=15) = 490314

term[15] := mul(490314, -1)

term[15] = -490314

choose 16 positions out of 23

C(23,16)=25852016738884976640000/(20922789888000·5040)=245157

x^7: (-1)^7

x^7 = -1

y^16: (-1)^16

y^16 = 1

C·x power (k=16) := mul(245157, -1)

C·x power (k=16) = -245157

term[16] := mul(-245157, 1)

term[16] = -245157

choose 17 positions out of 23

C(23,17)=25852016738884976640000/(355687428096000·720)=100947

x^6: (-1)^6

x^6 = 1

y^17: (-1)^17

y^17 = -1

C·x power (k=17) := mul(100947, 1)

C·x power (k=17) = 100947

term[17] := mul(100947, -1)

term[17] = -100947

choose 18 positions out of 23

C(23,18)=25852016738884976640000/(6402373705728000·120)=33649

x^5: (-1)^5

x^5 = -1

y^18: (-1)^18

y^18 = 1

C·x power (k=18) := mul(33649, -1)

C·x power (k=18) = -33649

term[18] := mul(-33649, 1)

term[18] = -33649

choose 19 positions out of 23

C(23,19)=25852016738884976640000/(121645100408832000·24)=8855

x^4: (-1)^4

x^4 = 1

y^19: (-1)^19

y^19 = -1

C·x power (k=19) := mul(8855, 1)

C·x power (k=19) = 8855

term[19] := mul(8855, -1)

term[19] = -8855

choose 20 positions out of 23

C(23,20)=25852016738884976640000/(2432902008176640000·6)=1771

x^3: (-1)^3

x^3 = -1

y^20: (-1)^20

y^20 = 1

C·x power (k=20) := mul(1771, -1)

C·x power (k=20) = -1771

term[20] := mul(-1771, 1)

term[20] = -1771

choose 21 positions out of 23

C(23,21)=25852016738884976640000/(51090942171709440000·2)=253

x^2: (-1)^2

x^2 = 1

y^21: (-1)^21

y^21 = -1

C·x power (k=21) := mul(253, 1)

C·x power (k=21) = 253

term[21] := mul(253, -1)

term[21] = -253

choose 22 positions out of 23

C(23,22)=25852016738884976640000/(1124000727777607680000·1)=23

x^1: (-1)^1

x^1 = -1

y^22: (-1)^22

y^22 = 1

C·x power (k=22) := mul(23, -1)

C·x power (k=22) = -23

term[22] := mul(-23, 1)

term[22] = -23

choose 23 positions out of 23

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

x^0: (-1)^0

x^0 = 1

y^23: (-1)^23

y^23 = -1

C·x power (k=23) := mul(1, 1)

C·x power (k=23) = 1

term[23] := mul(1, -1)

term[23] = -1

expanded sum := sum(-1, -23, -253, -1771, -8855, -33649, -100947, -245157, -490314, -817190, -1144066, -1352078, -1352078, -1144066, -817190, -490314, -245157, -100947, -33649, -8855, -1771, -253, -23, -1)

expanded sum = -8388608

direct=-8388608; expanded=-8388608

identity holds: Yes

Reverse checks

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

(23, -1, -1, -8388608, -8388608, 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.