Question 447: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=21, exact commuting integers x=-3, y=1. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=21, exact commuting integers x=-3, 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=21≥0; multiplication commutes

group the finite product by the number of y choices

x+y := sum(-3, 1)

x+y = -2

(x+y)^n: (-2)^21

(x+y)^n = -2097152

choose 0 positions out of 21

C(21,0)=51090942171709440000/(1·51090942171709440000)=1

x^21: (-3)^21

x^21 = -10460353203

y^0: (1)^0

y^0 = 1

C·x power (k=0) := mul(1, -10460353203)

C·x power (k=0) = -10460353203

term[0] := mul(-10460353203, 1)

term[0] = -10460353203

choose 1 positions out of 21

C(21,1)=51090942171709440000/(1·2432902008176640000)=21

x^20: (-3)^20

x^20 = 3486784401

y^1: (1)^1

y^1 = 1

C·x power (k=1) := mul(21, 3486784401)

C·x power (k=1) = 73222472421

term[1] := mul(73222472421, 1)

term[1] = 73222472421

choose 2 positions out of 21

C(21,2)=51090942171709440000/(2·121645100408832000)=210

x^19: (-3)^19

x^19 = -1162261467

y^2: (1)^2

y^2 = 1

C·x power (k=2) := mul(210, -1162261467)

C·x power (k=2) = -244074908070

term[2] := mul(-244074908070, 1)

term[2] = -244074908070

choose 3 positions out of 21

C(21,3)=51090942171709440000/(6·6402373705728000)=1330

x^18: (-3)^18

x^18 = 387420489

y^3: (1)^3

y^3 = 1

C·x power (k=3) := mul(1330, 387420489)

C·x power (k=3) = 515269250370

term[3] := mul(515269250370, 1)

term[3] = 515269250370

choose 4 positions out of 21

C(21,4)=51090942171709440000/(24·355687428096000)=5985

x^17: (-3)^17

x^17 = -129140163

y^4: (1)^4

y^4 = 1

C·x power (k=4) := mul(5985, -129140163)

C·x power (k=4) = -772903875555

term[4] := mul(-772903875555, 1)

term[4] = -772903875555

choose 5 positions out of 21

C(21,5)=51090942171709440000/(120·20922789888000)=20349

x^16: (-3)^16

x^16 = 43046721

y^5: (1)^5

y^5 = 1

C·x power (k=5) := mul(20349, 43046721)

C·x power (k=5) = 875957725629

term[5] := mul(875957725629, 1)

term[5] = 875957725629

choose 6 positions out of 21

C(21,6)=51090942171709440000/(720·1307674368000)=54264

x^15: (-3)^15

x^15 = -14348907

y^6: (1)^6

y^6 = 1

C·x power (k=6) := mul(54264, -14348907)

C·x power (k=6) = -778629089448

term[6] := mul(-778629089448, 1)

term[6] = -778629089448

choose 7 positions out of 21

C(21,7)=51090942171709440000/(5040·87178291200)=116280

x^14: (-3)^14

x^14 = 4782969

y^7: (1)^7

y^7 = 1

C·x power (k=7) := mul(116280, 4782969)

C·x power (k=7) = 556163635320

term[7] := mul(556163635320, 1)

term[7] = 556163635320

choose 8 positions out of 21

C(21,8)=51090942171709440000/(40320·6227020800)=203490

x^13: (-3)^13

x^13 = -1594323

y^8: (1)^8

y^8 = 1

C·x power (k=8) := mul(203490, -1594323)

C·x power (k=8) = -324428787270

term[8] := mul(-324428787270, 1)

term[8] = -324428787270

choose 9 positions out of 21

C(21,9)=51090942171709440000/(362880·479001600)=293930

x^12: (-3)^12

x^12 = 531441

y^9: (1)^9

y^9 = 1

C·x power (k=9) := mul(293930, 531441)

C·x power (k=9) = 156206453130

term[9] := mul(156206453130, 1)

term[9] = 156206453130

choose 10 positions out of 21

C(21,10)=51090942171709440000/(3628800·39916800)=352716

x^11: (-3)^11

x^11 = -177147

y^10: (1)^10

y^10 = 1

C·x power (k=10) := mul(352716, -177147)

C·x power (k=10) = -62482581252

term[10] := mul(-62482581252, 1)

term[10] = -62482581252

choose 11 positions out of 21

C(21,11)=51090942171709440000/(39916800·3628800)=352716

x^10: (-3)^10

x^10 = 59049

y^11: (1)^11

y^11 = 1

C·x power (k=11) := mul(352716, 59049)

C·x power (k=11) = 20827527084

term[11] := mul(20827527084, 1)

term[11] = 20827527084

choose 12 positions out of 21

C(21,12)=51090942171709440000/(479001600·362880)=293930

x^9: (-3)^9

x^9 = -19683

y^12: (1)^12

y^12 = 1

C·x power (k=12) := mul(293930, -19683)

C·x power (k=12) = -5785424190

term[12] := mul(-5785424190, 1)

term[12] = -5785424190

choose 13 positions out of 21

C(21,13)=51090942171709440000/(6227020800·40320)=203490

x^8: (-3)^8

x^8 = 6561

y^13: (1)^13

y^13 = 1

C·x power (k=13) := mul(203490, 6561)

C·x power (k=13) = 1335097890

term[13] := mul(1335097890, 1)

term[13] = 1335097890

choose 14 positions out of 21

C(21,14)=51090942171709440000/(87178291200·5040)=116280

x^7: (-3)^7

x^7 = -2187

y^14: (1)^14

y^14 = 1

C·x power (k=14) := mul(116280, -2187)

C·x power (k=14) = -254304360

term[14] := mul(-254304360, 1)

term[14] = -254304360

choose 15 positions out of 21

C(21,15)=51090942171709440000/(1307674368000·720)=54264

x^6: (-3)^6

x^6 = 729

y^15: (1)^15

y^15 = 1

C·x power (k=15) := mul(54264, 729)

C·x power (k=15) = 39558456

term[15] := mul(39558456, 1)

term[15] = 39558456

choose 16 positions out of 21

C(21,16)=51090942171709440000/(20922789888000·120)=20349

x^5: (-3)^5

x^5 = -243

y^16: (1)^16

y^16 = 1

C·x power (k=16) := mul(20349, -243)

C·x power (k=16) = -4944807

term[16] := mul(-4944807, 1)

term[16] = -4944807

choose 17 positions out of 21

C(21,17)=51090942171709440000/(355687428096000·24)=5985

x^4: (-3)^4

x^4 = 81

y^17: (1)^17

y^17 = 1

C·x power (k=17) := mul(5985, 81)

C·x power (k=17) = 484785

term[17] := mul(484785, 1)

term[17] = 484785

choose 18 positions out of 21

C(21,18)=51090942171709440000/(6402373705728000·6)=1330

x^3: (-3)^3

x^3 = -27

y^18: (1)^18

y^18 = 1

C·x power (k=18) := mul(1330, -27)

C·x power (k=18) = -35910

term[18] := mul(-35910, 1)

term[18] = -35910

choose 19 positions out of 21

C(21,19)=51090942171709440000/(121645100408832000·2)=210

x^2: (-3)^2

x^2 = 9

y^19: (1)^19

y^19 = 1

C·x power (k=19) := mul(210, 9)

C·x power (k=19) = 1890

term[19] := mul(1890, 1)

term[19] = 1890

choose 20 positions out of 21

C(21,20)=51090942171709440000/(2432902008176640000·1)=21

x^1: (-3)^1

x^1 = -3

y^20: (1)^20

y^20 = 1

C·x power (k=20) := mul(21, -3)

C·x power (k=20) = -63

term[20] := mul(-63, 1)

term[20] = -63

choose 21 positions out of 21

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

x^0: (-3)^0

x^0 = 1

y^21: (1)^21

y^21 = 1

C·x power (k=21) := mul(1, 1)

C·x power (k=21) = 1

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

term[21] = 1

expanded sum := sum(-10460353203, 73222472421, -244074908070, 515269250370, -772903875555, 875957725629, -778629089448, 556163635320, -324428787270, 156206453130, -62482581252, 20827527084, -5785424190, 1335097890, -254304360, 39558456, -4944807, 484785, -35910, 1890, -63, 1)

expanded sum = -2097152

direct=-2097152; expanded=-2097152

identity holds: Yes

Reverse checks

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

(21, -3, 1, -2097152, -2097152, 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.