Question 431: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=5, exact commuting integers x=2, y=2. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=5, 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=5≥0; multiplication commutes

group the finite product by the number of y choices

x+y := sum(2, 2)

x+y = 4

(x+y)^n: (4)^5

(x+y)^n = 1024

choose 0 positions out of 5

C(5,0)=120/(1·120)=1

x^5: (2)^5

x^5 = 32

y^0: (2)^0

y^0 = 1

C·x power (k=0) := mul(1, 32)

C·x power (k=0) = 32

term[0] := mul(32, 1)

term[0] = 32

choose 1 positions out of 5

C(5,1)=120/(1·24)=5

x^4: (2)^4

x^4 = 16

y^1: (2)^1

y^1 = 2

C·x power (k=1) := mul(5, 16)

C·x power (k=1) = 80

term[1] := mul(80, 2)

term[1] = 160

choose 2 positions out of 5

C(5,2)=120/(2·6)=10

x^3: (2)^3

x^3 = 8

y^2: (2)^2

y^2 = 4

C·x power (k=2) := mul(10, 8)

C·x power (k=2) = 80

term[2] := mul(80, 4)

term[2] = 320

choose 3 positions out of 5

C(5,3)=120/(6·2)=10

x^2: (2)^2

x^2 = 4

y^3: (2)^3

y^3 = 8

C·x power (k=3) := mul(10, 4)

C·x power (k=3) = 40

term[3] := mul(40, 8)

term[3] = 320

choose 4 positions out of 5

C(5,4)=120/(24·1)=5

x^1: (2)^1

x^1 = 2

y^4: (2)^4

y^4 = 16

C·x power (k=4) := mul(5, 2)

C·x power (k=4) = 10

term[4] := mul(10, 16)

term[4] = 160

choose 5 positions out of 5

C(5,5)=120/(120·1)=1

x^0: (2)^0

x^0 = 1

y^5: (2)^5

y^5 = 32

C·x power (k=5) := mul(1, 1)

C·x power (k=5) = 1

term[5] := mul(1, 32)

term[5] = 32

expanded sum := sum(32, 160, 320, 320, 160, 32)

expanded sum = 1024

direct=1024; expanded=1024

identity holds: Yes

Reverse checks

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

(5, 2, 2, 1024, 1024, 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.