Question 433: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=7, exact commuting integers x=-3, y=0. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=7, exact commuting integers x=-3, 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=7≥0; multiplication commutes

group the finite product by the number of y choices

x+y := sum(-3, 0)

x+y = -3

(x+y)^n: (-3)^7

(x+y)^n = -2187

choose 0 positions out of 7

C(7,0)=5040/(1·5040)=1

x^7: (-3)^7

x^7 = -2187

y^0: (0)^0

y^0 = 1

C·x power (k=0) := mul(1, -2187)

C·x power (k=0) = -2187

term[0] := mul(-2187, 1)

term[0] = -2187

choose 1 positions out of 7

C(7,1)=5040/(1·720)=7

x^6: (-3)^6

x^6 = 729

y^1: (0)^1

y^1 = 0

C·x power (k=1) := mul(7, 729)

C·x power (k=1) = 5103

term[1] := mul(5103, 0)

term[1] = 0

choose 2 positions out of 7

C(7,2)=5040/(2·120)=21

x^5: (-3)^5

x^5 = -243

y^2: (0)^2

y^2 = 0

C·x power (k=2) := mul(21, -243)

C·x power (k=2) = -5103

term[2] := mul(-5103, 0)

term[2] = 0

choose 3 positions out of 7

C(7,3)=5040/(6·24)=35

x^4: (-3)^4

x^4 = 81

y^3: (0)^3

y^3 = 0

C·x power (k=3) := mul(35, 81)

C·x power (k=3) = 2835

term[3] := mul(2835, 0)

term[3] = 0

choose 4 positions out of 7

C(7,4)=5040/(24·6)=35

x^3: (-3)^3

x^3 = -27

y^4: (0)^4

y^4 = 0

C·x power (k=4) := mul(35, -27)

C·x power (k=4) = -945

term[4] := mul(-945, 0)

term[4] = 0

choose 5 positions out of 7

C(7,5)=5040/(120·2)=21

x^2: (-3)^2

x^2 = 9

y^5: (0)^5

y^5 = 0

C·x power (k=5) := mul(21, 9)

C·x power (k=5) = 189

term[5] := mul(189, 0)

term[5] = 0

choose 6 positions out of 7

C(7,6)=5040/(720·1)=7

x^1: (-3)^1

x^1 = -3

y^6: (0)^6

y^6 = 0

C·x power (k=6) := mul(7, -3)

C·x power (k=6) = -21

term[6] := mul(-21, 0)

term[6] = 0

choose 7 positions out of 7

C(7,7)=5040/(5040·1)=1

x^0: (-3)^0

x^0 = 1

y^7: (0)^7

y^7 = 0

C·x power (k=7) := mul(1, 1)

C·x power (k=7) = 1

term[7] := mul(1, 0)

term[7] = 0

expanded sum := sum(-2187, 0, 0, 0, 0, 0, 0, 0)

expanded sum = -2187

direct=-2187; expanded=-2187

identity holds: Yes

Reverse checks

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

(7, -3, 0, -2187, -2187, 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.