Question 435: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=9, exact commuting integers x=-1, y=-2. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=9, exact commuting integers x=-1, 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=9≥0; multiplication commutes

group the finite product by the number of y choices

x+y := sum(-1, -2)

x+y = -3

(x+y)^n: (-3)^9

(x+y)^n = -19683

choose 0 positions out of 9

C(9,0)=362880/(1·362880)=1

x^9: (-1)^9

x^9 = -1

y^0: (-2)^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 9

C(9,1)=362880/(1·40320)=9

x^8: (-1)^8

x^8 = 1

y^1: (-2)^1

y^1 = -2

C·x power (k=1) := mul(9, 1)

C·x power (k=1) = 9

term[1] := mul(9, -2)

term[1] = -18

choose 2 positions out of 9

C(9,2)=362880/(2·5040)=36

x^7: (-1)^7

x^7 = -1

y^2: (-2)^2

y^2 = 4

C·x power (k=2) := mul(36, -1)

C·x power (k=2) = -36

term[2] := mul(-36, 4)

term[2] = -144

choose 3 positions out of 9

C(9,3)=362880/(6·720)=84

x^6: (-1)^6

x^6 = 1

y^3: (-2)^3

y^3 = -8

C·x power (k=3) := mul(84, 1)

C·x power (k=3) = 84

term[3] := mul(84, -8)

term[3] = -672

choose 4 positions out of 9

C(9,4)=362880/(24·120)=126

x^5: (-1)^5

x^5 = -1

y^4: (-2)^4

y^4 = 16

C·x power (k=4) := mul(126, -1)

C·x power (k=4) = -126

term[4] := mul(-126, 16)

term[4] = -2016

choose 5 positions out of 9

C(9,5)=362880/(120·24)=126

x^4: (-1)^4

x^4 = 1

y^5: (-2)^5

y^5 = -32

C·x power (k=5) := mul(126, 1)

C·x power (k=5) = 126

term[5] := mul(126, -32)

term[5] = -4032

choose 6 positions out of 9

C(9,6)=362880/(720·6)=84

x^3: (-1)^3

x^3 = -1

y^6: (-2)^6

y^6 = 64

C·x power (k=6) := mul(84, -1)

C·x power (k=6) = -84

term[6] := mul(-84, 64)

term[6] = -5376

choose 7 positions out of 9

C(9,7)=362880/(5040·2)=36

x^2: (-1)^2

x^2 = 1

y^7: (-2)^7

y^7 = -128

C·x power (k=7) := mul(36, 1)

C·x power (k=7) = 36

term[7] := mul(36, -128)

term[7] = -4608

choose 8 positions out of 9

C(9,8)=362880/(40320·1)=9

x^1: (-1)^1

x^1 = -1

y^8: (-2)^8

y^8 = 256

C·x power (k=8) := mul(9, -1)

C·x power (k=8) = -9

term[8] := mul(-9, 256)

term[8] = -2304

choose 9 positions out of 9

C(9,9)=362880/(362880·1)=1

x^0: (-1)^0

x^0 = 1

y^9: (-2)^9

y^9 = -512

C·x power (k=9) := mul(1, 1)

C·x power (k=9) = 1

term[9] := mul(1, -512)

term[9] = -512

expanded sum := sum(-1, -18, -144, -672, -2016, -4032, -5376, -4608, -2304, -512)

expanded sum = -19683

direct=-19683; expanded=-19683

identity holds: Yes

Reverse checks

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

(9, -1, -2, -19683, -19683, 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.