Question 439: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=13, exact commuting integers x=3, y=-1. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=13, exact commuting integers x=3, 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=13≥0; multiplication commutes

group the finite product by the number of y choices

x+y := sum(3, -1)

x+y = 2

(x+y)^n: (2)^13

(x+y)^n = 8192

choose 0 positions out of 13

C(13,0)=6227020800/(1·6227020800)=1

x^13: (3)^13

x^13 = 1594323

y^0: (-1)^0

y^0 = 1

C·x power (k=0) := mul(1, 1594323)

C·x power (k=0) = 1594323

term[0] := mul(1594323, 1)

term[0] = 1594323

choose 1 positions out of 13

C(13,1)=6227020800/(1·479001600)=13

x^12: (3)^12

x^12 = 531441

y^1: (-1)^1

y^1 = -1

C·x power (k=1) := mul(13, 531441)

C·x power (k=1) = 6908733

term[1] := mul(6908733, -1)

term[1] = -6908733

choose 2 positions out of 13

C(13,2)=6227020800/(2·39916800)=78

x^11: (3)^11

x^11 = 177147

y^2: (-1)^2

y^2 = 1

C·x power (k=2) := mul(78, 177147)

C·x power (k=2) = 13817466

term[2] := mul(13817466, 1)

term[2] = 13817466

choose 3 positions out of 13

C(13,3)=6227020800/(6·3628800)=286

x^10: (3)^10

x^10 = 59049

y^3: (-1)^3

y^3 = -1

C·x power (k=3) := mul(286, 59049)

C·x power (k=3) = 16888014

term[3] := mul(16888014, -1)

term[3] = -16888014

choose 4 positions out of 13

C(13,4)=6227020800/(24·362880)=715

x^9: (3)^9

x^9 = 19683

y^4: (-1)^4

y^4 = 1

C·x power (k=4) := mul(715, 19683)

C·x power (k=4) = 14073345

term[4] := mul(14073345, 1)

term[4] = 14073345

choose 5 positions out of 13

C(13,5)=6227020800/(120·40320)=1287

x^8: (3)^8

x^8 = 6561

y^5: (-1)^5

y^5 = -1

C·x power (k=5) := mul(1287, 6561)

C·x power (k=5) = 8444007

term[5] := mul(8444007, -1)

term[5] = -8444007

choose 6 positions out of 13

C(13,6)=6227020800/(720·5040)=1716

x^7: (3)^7

x^7 = 2187

y^6: (-1)^6

y^6 = 1

C·x power (k=6) := mul(1716, 2187)

C·x power (k=6) = 3752892

term[6] := mul(3752892, 1)

term[6] = 3752892

choose 7 positions out of 13

C(13,7)=6227020800/(5040·720)=1716

x^6: (3)^6

x^6 = 729

y^7: (-1)^7

y^7 = -1

C·x power (k=7) := mul(1716, 729)

C·x power (k=7) = 1250964

term[7] := mul(1250964, -1)

term[7] = -1250964

choose 8 positions out of 13

C(13,8)=6227020800/(40320·120)=1287

x^5: (3)^5

x^5 = 243

y^8: (-1)^8

y^8 = 1

C·x power (k=8) := mul(1287, 243)

C·x power (k=8) = 312741

term[8] := mul(312741, 1)

term[8] = 312741

choose 9 positions out of 13

C(13,9)=6227020800/(362880·24)=715

x^4: (3)^4

x^4 = 81

y^9: (-1)^9

y^9 = -1

C·x power (k=9) := mul(715, 81)

C·x power (k=9) = 57915

term[9] := mul(57915, -1)

term[9] = -57915

choose 10 positions out of 13

C(13,10)=6227020800/(3628800·6)=286

x^3: (3)^3

x^3 = 27

y^10: (-1)^10

y^10 = 1

C·x power (k=10) := mul(286, 27)

C·x power (k=10) = 7722

term[10] := mul(7722, 1)

term[10] = 7722

choose 11 positions out of 13

C(13,11)=6227020800/(39916800·2)=78

x^2: (3)^2

x^2 = 9

y^11: (-1)^11

y^11 = -1

C·x power (k=11) := mul(78, 9)

C·x power (k=11) = 702

term[11] := mul(702, -1)

term[11] = -702

choose 12 positions out of 13

C(13,12)=6227020800/(479001600·1)=13

x^1: (3)^1

x^1 = 3

y^12: (-1)^12

y^12 = 1

C·x power (k=12) := mul(13, 3)

C·x power (k=12) = 39

term[12] := mul(39, 1)

term[12] = 39

choose 13 positions out of 13

C(13,13)=6227020800/(6227020800·1)=1

x^0: (3)^0

x^0 = 1

y^13: (-1)^13

y^13 = -1

C·x power (k=13) := mul(1, 1)

C·x power (k=13) = 1

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

term[13] = -1

expanded sum := sum(1594323, -6908733, 13817466, -16888014, 14073345, -8444007, 3752892, -1250964, 312741, -57915, 7722, -702, 39, -1)

expanded sum = 8192

direct=8192; expanded=8192

identity holds: Yes

Reverse checks

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

(13, 3, -1, 8192, 8192, 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.