Question 442: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=16, exact commuting integers x=-1, y=1. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=16, exact commuting integers x=-1, 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=16≥0; multiplication commutes

group the finite product by the number of y choices

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

x+y = 0

(x+y)^n: (0)^16

(x+y)^n = 0

choose 0 positions out of 16

C(16,0)=20922789888000/(1·20922789888000)=1

x^16: (-1)^16

x^16 = 1

y^0: (1)^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 16

C(16,1)=20922789888000/(1·1307674368000)=16

x^15: (-1)^15

x^15 = -1

y^1: (1)^1

y^1 = 1

C·x power (k=1) := mul(16, -1)

C·x power (k=1) = -16

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

term[1] = -16

choose 2 positions out of 16

C(16,2)=20922789888000/(2·87178291200)=120

x^14: (-1)^14

x^14 = 1

y^2: (1)^2

y^2 = 1

C·x power (k=2) := mul(120, 1)

C·x power (k=2) = 120

term[2] := mul(120, 1)

term[2] = 120

choose 3 positions out of 16

C(16,3)=20922789888000/(6·6227020800)=560

x^13: (-1)^13

x^13 = -1

y^3: (1)^3

y^3 = 1

C·x power (k=3) := mul(560, -1)

C·x power (k=3) = -560

term[3] := mul(-560, 1)

term[3] = -560

choose 4 positions out of 16

C(16,4)=20922789888000/(24·479001600)=1820

x^12: (-1)^12

x^12 = 1

y^4: (1)^4

y^4 = 1

C·x power (k=4) := mul(1820, 1)

C·x power (k=4) = 1820

term[4] := mul(1820, 1)

term[4] = 1820

choose 5 positions out of 16

C(16,5)=20922789888000/(120·39916800)=4368

x^11: (-1)^11

x^11 = -1

y^5: (1)^5

y^5 = 1

C·x power (k=5) := mul(4368, -1)

C·x power (k=5) = -4368

term[5] := mul(-4368, 1)

term[5] = -4368

choose 6 positions out of 16

C(16,6)=20922789888000/(720·3628800)=8008

x^10: (-1)^10

x^10 = 1

y^6: (1)^6

y^6 = 1

C·x power (k=6) := mul(8008, 1)

C·x power (k=6) = 8008

term[6] := mul(8008, 1)

term[6] = 8008

choose 7 positions out of 16

C(16,7)=20922789888000/(5040·362880)=11440

x^9: (-1)^9

x^9 = -1

y^7: (1)^7

y^7 = 1

C·x power (k=7) := mul(11440, -1)

C·x power (k=7) = -11440

term[7] := mul(-11440, 1)

term[7] = -11440

choose 8 positions out of 16

C(16,8)=20922789888000/(40320·40320)=12870

x^8: (-1)^8

x^8 = 1

y^8: (1)^8

y^8 = 1

C·x power (k=8) := mul(12870, 1)

C·x power (k=8) = 12870

term[8] := mul(12870, 1)

term[8] = 12870

choose 9 positions out of 16

C(16,9)=20922789888000/(362880·5040)=11440

x^7: (-1)^7

x^7 = -1

y^9: (1)^9

y^9 = 1

C·x power (k=9) := mul(11440, -1)

C·x power (k=9) = -11440

term[9] := mul(-11440, 1)

term[9] = -11440

choose 10 positions out of 16

C(16,10)=20922789888000/(3628800·720)=8008

x^6: (-1)^6

x^6 = 1

y^10: (1)^10

y^10 = 1

C·x power (k=10) := mul(8008, 1)

C·x power (k=10) = 8008

term[10] := mul(8008, 1)

term[10] = 8008

choose 11 positions out of 16

C(16,11)=20922789888000/(39916800·120)=4368

x^5: (-1)^5

x^5 = -1

y^11: (1)^11

y^11 = 1

C·x power (k=11) := mul(4368, -1)

C·x power (k=11) = -4368

term[11] := mul(-4368, 1)

term[11] = -4368

choose 12 positions out of 16

C(16,12)=20922789888000/(479001600·24)=1820

x^4: (-1)^4

x^4 = 1

y^12: (1)^12

y^12 = 1

C·x power (k=12) := mul(1820, 1)

C·x power (k=12) = 1820

term[12] := mul(1820, 1)

term[12] = 1820

choose 13 positions out of 16

C(16,13)=20922789888000/(6227020800·6)=560

x^3: (-1)^3

x^3 = -1

y^13: (1)^13

y^13 = 1

C·x power (k=13) := mul(560, -1)

C·x power (k=13) = -560

term[13] := mul(-560, 1)

term[13] = -560

choose 14 positions out of 16

C(16,14)=20922789888000/(87178291200·2)=120

x^2: (-1)^2

x^2 = 1

y^14: (1)^14

y^14 = 1

C·x power (k=14) := mul(120, 1)

C·x power (k=14) = 120

term[14] := mul(120, 1)

term[14] = 120

choose 15 positions out of 16

C(16,15)=20922789888000/(1307674368000·1)=16

x^1: (-1)^1

x^1 = -1

y^15: (1)^15

y^15 = 1

C·x power (k=15) := mul(16, -1)

C·x power (k=15) = -16

term[15] := mul(-16, 1)

term[15] = -16

choose 16 positions out of 16

C(16,16)=20922789888000/(20922789888000·1)=1

x^0: (-1)^0

x^0 = 1

y^16: (1)^16

y^16 = 1

C·x power (k=16) := mul(1, 1)

C·x power (k=16) = 1

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

term[16] = 1

expanded sum := sum(1, -16, 120, -560, 1820, -4368, 8008, -11440, 12870, -11440, 8008, -4368, 1820, -560, 120, -16, 1)

expanded sum = 0

direct=0; expanded=0

identity holds: Yes

Reverse checks

• S090 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S089 · premise_and_result_audit — Recompute expanded sum with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute term[16] with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute C·x power (k=16) with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S085 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S084 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S083 · premise_and_result_audit — Recompute term[15] with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute C·x power (k=15) with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S080 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S079 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S078 · premise_and_result_audit — Recompute term[14] with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute C·x power (k=14) with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S075 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S074 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S073 · premise_and_result_audit — Recompute term[13] with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute C·x power (k=13) with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S070 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S069 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S068 · premise_and_result_audit — Recompute term[12] with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute C·x power (k=12) with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S065 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S064 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S063 · premise_and_result_audit — Recompute term[11] with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute C·x power (k=11) with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S060 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S059 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S058 · premise_and_result_audit — Recompute term[10] with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute C·x power (k=10) with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• 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 — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• 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

(16, -1, 1, 0, 0, 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.