Question 441: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=15, exact commuting integers x=-2, y=2. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=15, 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=15≥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)^15

(x+y)^n = 0

choose 0 positions out of 15

C(15,0)=1307674368000/(1·1307674368000)=1

x^15: (-2)^15

x^15 = -32768

y^0: (2)^0

y^0 = 1

C·x power (k=0) := mul(1, -32768)

C·x power (k=0) = -32768

term[0] := mul(-32768, 1)

term[0] = -32768

choose 1 positions out of 15

C(15,1)=1307674368000/(1·87178291200)=15

x^14: (-2)^14

x^14 = 16384

y^1: (2)^1

y^1 = 2

C·x power (k=1) := mul(15, 16384)

C·x power (k=1) = 245760

term[1] := mul(245760, 2)

term[1] = 491520

choose 2 positions out of 15

C(15,2)=1307674368000/(2·6227020800)=105

x^13: (-2)^13

x^13 = -8192

y^2: (2)^2

y^2 = 4

C·x power (k=2) := mul(105, -8192)

C·x power (k=2) = -860160

term[2] := mul(-860160, 4)

term[2] = -3440640

choose 3 positions out of 15

C(15,3)=1307674368000/(6·479001600)=455

x^12: (-2)^12

x^12 = 4096

y^3: (2)^3

y^3 = 8

C·x power (k=3) := mul(455, 4096)

C·x power (k=3) = 1863680

term[3] := mul(1863680, 8)

term[3] = 14909440

choose 4 positions out of 15

C(15,4)=1307674368000/(24·39916800)=1365

x^11: (-2)^11

x^11 = -2048

y^4: (2)^4

y^4 = 16

C·x power (k=4) := mul(1365, -2048)

C·x power (k=4) = -2795520

term[4] := mul(-2795520, 16)

term[4] = -44728320

choose 5 positions out of 15

C(15,5)=1307674368000/(120·3628800)=3003

x^10: (-2)^10

x^10 = 1024

y^5: (2)^5

y^5 = 32

C·x power (k=5) := mul(3003, 1024)

C·x power (k=5) = 3075072

term[5] := mul(3075072, 32)

term[5] = 98402304

choose 6 positions out of 15

C(15,6)=1307674368000/(720·362880)=5005

x^9: (-2)^9

x^9 = -512

y^6: (2)^6

y^6 = 64

C·x power (k=6) := mul(5005, -512)

C·x power (k=6) = -2562560

term[6] := mul(-2562560, 64)

term[6] = -164003840

choose 7 positions out of 15

C(15,7)=1307674368000/(5040·40320)=6435

x^8: (-2)^8

x^8 = 256

y^7: (2)^7

y^7 = 128

C·x power (k=7) := mul(6435, 256)

C·x power (k=7) = 1647360

term[7] := mul(1647360, 128)

term[7] = 210862080

choose 8 positions out of 15

C(15,8)=1307674368000/(40320·5040)=6435

x^7: (-2)^7

x^7 = -128

y^8: (2)^8

y^8 = 256

C·x power (k=8) := mul(6435, -128)

C·x power (k=8) = -823680

term[8] := mul(-823680, 256)

term[8] = -210862080

choose 9 positions out of 15

C(15,9)=1307674368000/(362880·720)=5005

x^6: (-2)^6

x^6 = 64

y^9: (2)^9

y^9 = 512

C·x power (k=9) := mul(5005, 64)

C·x power (k=9) = 320320

term[9] := mul(320320, 512)

term[9] = 164003840

choose 10 positions out of 15

C(15,10)=1307674368000/(3628800·120)=3003

x^5: (-2)^5

x^5 = -32

y^10: (2)^10

y^10 = 1024

C·x power (k=10) := mul(3003, -32)

C·x power (k=10) = -96096

term[10] := mul(-96096, 1024)

term[10] = -98402304

choose 11 positions out of 15

C(15,11)=1307674368000/(39916800·24)=1365

x^4: (-2)^4

x^4 = 16

y^11: (2)^11

y^11 = 2048

C·x power (k=11) := mul(1365, 16)

C·x power (k=11) = 21840

term[11] := mul(21840, 2048)

term[11] = 44728320

choose 12 positions out of 15

C(15,12)=1307674368000/(479001600·6)=455

x^3: (-2)^3

x^3 = -8

y^12: (2)^12

y^12 = 4096

C·x power (k=12) := mul(455, -8)

C·x power (k=12) = -3640

term[12] := mul(-3640, 4096)

term[12] = -14909440

choose 13 positions out of 15

C(15,13)=1307674368000/(6227020800·2)=105

x^2: (-2)^2

x^2 = 4

y^13: (2)^13

y^13 = 8192

C·x power (k=13) := mul(105, 4)

C·x power (k=13) = 420

term[13] := mul(420, 8192)

term[13] = 3440640

choose 14 positions out of 15

C(15,14)=1307674368000/(87178291200·1)=15

x^1: (-2)^1

x^1 = -2

y^14: (2)^14

y^14 = 16384

C·x power (k=14) := mul(15, -2)

C·x power (k=14) = -30

term[14] := mul(-30, 16384)

term[14] = -491520

choose 15 positions out of 15

C(15,15)=1307674368000/(1307674368000·1)=1

x^0: (-2)^0

x^0 = 1

y^15: (2)^15

y^15 = 32768

C·x power (k=15) := mul(1, 1)

C·x power (k=15) = 1

term[15] := mul(1, 32768)

term[15] = 32768

expanded sum := sum(-32768, 491520, -3440640, 14909440, -44728320, 98402304, -164003840, 210862080, -210862080, 164003840, -98402304, 44728320, -14909440, 3440640, -491520, 32768)

expanded sum = 0

direct=0; expanded=0

identity holds: Yes

Reverse checks

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

(15, -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.