Question 434: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=8, exact commuting integers x=-2, y=-1. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=8, exact commuting integers x=-2, 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=8≥0; multiplication commutes

group the finite product by the number of y choices

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

x+y = -3

(x+y)^n: (-3)^8

(x+y)^n = 6561

choose 0 positions out of 8

C(8,0)=40320/(1·40320)=1

x^8: (-2)^8

x^8 = 256

y^0: (-1)^0

y^0 = 1

C·x power (k=0) := mul(1, 256)

C·x power (k=0) = 256

term[0] := mul(256, 1)

term[0] = 256

choose 1 positions out of 8

C(8,1)=40320/(1·5040)=8

x^7: (-2)^7

x^7 = -128

y^1: (-1)^1

y^1 = -1

C·x power (k=1) := mul(8, -128)

C·x power (k=1) = -1024

term[1] := mul(-1024, -1)

term[1] = 1024

choose 2 positions out of 8

C(8,2)=40320/(2·720)=28

x^6: (-2)^6

x^6 = 64

y^2: (-1)^2

y^2 = 1

C·x power (k=2) := mul(28, 64)

C·x power (k=2) = 1792

term[2] := mul(1792, 1)

term[2] = 1792

choose 3 positions out of 8

C(8,3)=40320/(6·120)=56

x^5: (-2)^5

x^5 = -32

y^3: (-1)^3

y^3 = -1

C·x power (k=3) := mul(56, -32)

C·x power (k=3) = -1792

term[3] := mul(-1792, -1)

term[3] = 1792

choose 4 positions out of 8

C(8,4)=40320/(24·24)=70

x^4: (-2)^4

x^4 = 16

y^4: (-1)^4

y^4 = 1

C·x power (k=4) := mul(70, 16)

C·x power (k=4) = 1120

term[4] := mul(1120, 1)

term[4] = 1120

choose 5 positions out of 8

C(8,5)=40320/(120·6)=56

x^3: (-2)^3

x^3 = -8

y^5: (-1)^5

y^5 = -1

C·x power (k=5) := mul(56, -8)

C·x power (k=5) = -448

term[5] := mul(-448, -1)

term[5] = 448

choose 6 positions out of 8

C(8,6)=40320/(720·2)=28

x^2: (-2)^2

x^2 = 4

y^6: (-1)^6

y^6 = 1

C·x power (k=6) := mul(28, 4)

C·x power (k=6) = 112

term[6] := mul(112, 1)

term[6] = 112

choose 7 positions out of 8

C(8,7)=40320/(5040·1)=8

x^1: (-2)^1

x^1 = -2

y^7: (-1)^7

y^7 = -1

C·x power (k=7) := mul(8, -2)

C·x power (k=7) = -16

term[7] := mul(-16, -1)

term[7] = 16

choose 8 positions out of 8

C(8,8)=40320/(40320·1)=1

x^0: (-2)^0

x^0 = 1

y^8: (-1)^8

y^8 = 1

C·x power (k=8) := mul(1, 1)

C·x power (k=8) = 1

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

term[8] = 1

expanded sum := sum(256, 1024, 1792, 1792, 1120, 448, 112, 16, 1)

expanded sum = 6561

direct=6561; expanded=6561

identity holds: Yes

Reverse checks

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

(8, -2, -1, 6561, 6561, 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.