Question 428: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=2, exact commuting integers x=-1, y=0. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=2, exact commuting integers x=-1, 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=2≥0; multiplication commutes

group the finite product by the number of y choices

x+y := sum(-1, 0)

x+y = -1

(x+y)^n: (-1)^2

(x+y)^n = 1

choose 0 positions out of 2

C(2,0)=2/(1·2)=1

x^2: (-1)^2

x^2 = 1

y^0: (0)^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 2

C(2,1)=2/(1·1)=2

x^1: (-1)^1

x^1 = -1

y^1: (0)^1

y^1 = 0

C·x power (k=1) := mul(2, -1)

C·x power (k=1) = -2

term[1] := mul(-2, 0)

term[1] = 0

choose 2 positions out of 2

C(2,2)=2/(2·1)=1

x^0: (-1)^0

x^0 = 1

y^2: (0)^2

y^2 = 0

C·x power (k=2) := mul(1, 1)

C·x power (k=2) = 1

term[2] := mul(1, 0)

term[2] = 0

expanded sum := sum(1, 0, 0)

expanded sum = 1

direct=1; expanded=1

identity holds: Yes

Reverse checks

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

(2, -1, 0, 1, 1, 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.