Question 440: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=14, exact commuting integers x=-3, y=-2. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=14, exact commuting integers x=-3, 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=14≥0; multiplication commutes

group the finite product by the number of y choices

x+y := sum(-3, -2)

x+y = -5

(x+y)^n: (-5)^14

(x+y)^n = 6103515625

choose 0 positions out of 14

C(14,0)=87178291200/(1·87178291200)=1

x^14: (-3)^14

x^14 = 4782969

y^0: (-2)^0

y^0 = 1

C·x power (k=0) := mul(1, 4782969)

C·x power (k=0) = 4782969

term[0] := mul(4782969, 1)

term[0] = 4782969

choose 1 positions out of 14

C(14,1)=87178291200/(1·6227020800)=14

x^13: (-3)^13

x^13 = -1594323

y^1: (-2)^1

y^1 = -2

C·x power (k=1) := mul(14, -1594323)

C·x power (k=1) = -22320522

term[1] := mul(-22320522, -2)

term[1] = 44641044

choose 2 positions out of 14

C(14,2)=87178291200/(2·479001600)=91

x^12: (-3)^12

x^12 = 531441

y^2: (-2)^2

y^2 = 4

C·x power (k=2) := mul(91, 531441)

C·x power (k=2) = 48361131

term[2] := mul(48361131, 4)

term[2] = 193444524

choose 3 positions out of 14

C(14,3)=87178291200/(6·39916800)=364

x^11: (-3)^11

x^11 = -177147

y^3: (-2)^3

y^3 = -8

C·x power (k=3) := mul(364, -177147)

C·x power (k=3) = -64481508

term[3] := mul(-64481508, -8)

term[3] = 515852064

choose 4 positions out of 14

C(14,4)=87178291200/(24·3628800)=1001

x^10: (-3)^10

x^10 = 59049

y^4: (-2)^4

y^4 = 16

C·x power (k=4) := mul(1001, 59049)

C·x power (k=4) = 59108049

term[4] := mul(59108049, 16)

term[4] = 945728784

choose 5 positions out of 14

C(14,5)=87178291200/(120·362880)=2002

x^9: (-3)^9

x^9 = -19683

y^5: (-2)^5

y^5 = -32

C·x power (k=5) := mul(2002, -19683)

C·x power (k=5) = -39405366

term[5] := mul(-39405366, -32)

term[5] = 1260971712

choose 6 positions out of 14

C(14,6)=87178291200/(720·40320)=3003

x^8: (-3)^8

x^8 = 6561

y^6: (-2)^6

y^6 = 64

C·x power (k=6) := mul(3003, 6561)

C·x power (k=6) = 19702683

term[6] := mul(19702683, 64)

term[6] = 1260971712

choose 7 positions out of 14

C(14,7)=87178291200/(5040·5040)=3432

x^7: (-3)^7

x^7 = -2187

y^7: (-2)^7

y^7 = -128

C·x power (k=7) := mul(3432, -2187)

C·x power (k=7) = -7505784

term[7] := mul(-7505784, -128)

term[7] = 960740352

choose 8 positions out of 14

C(14,8)=87178291200/(40320·720)=3003

x^6: (-3)^6

x^6 = 729

y^8: (-2)^8

y^8 = 256

C·x power (k=8) := mul(3003, 729)

C·x power (k=8) = 2189187

term[8] := mul(2189187, 256)

term[8] = 560431872

choose 9 positions out of 14

C(14,9)=87178291200/(362880·120)=2002

x^5: (-3)^5

x^5 = -243

y^9: (-2)^9

y^9 = -512

C·x power (k=9) := mul(2002, -243)

C·x power (k=9) = -486486

term[9] := mul(-486486, -512)

term[9] = 249080832

choose 10 positions out of 14

C(14,10)=87178291200/(3628800·24)=1001

x^4: (-3)^4

x^4 = 81

y^10: (-2)^10

y^10 = 1024

C·x power (k=10) := mul(1001, 81)

C·x power (k=10) = 81081

term[10] := mul(81081, 1024)

term[10] = 83026944

choose 11 positions out of 14

C(14,11)=87178291200/(39916800·6)=364

x^3: (-3)^3

x^3 = -27

y^11: (-2)^11

y^11 = -2048

C·x power (k=11) := mul(364, -27)

C·x power (k=11) = -9828

term[11] := mul(-9828, -2048)

term[11] = 20127744

choose 12 positions out of 14

C(14,12)=87178291200/(479001600·2)=91

x^2: (-3)^2

x^2 = 9

y^12: (-2)^12

y^12 = 4096

C·x power (k=12) := mul(91, 9)

C·x power (k=12) = 819

term[12] := mul(819, 4096)

term[12] = 3354624

choose 13 positions out of 14

C(14,13)=87178291200/(6227020800·1)=14

x^1: (-3)^1

x^1 = -3

y^13: (-2)^13

y^13 = -8192

C·x power (k=13) := mul(14, -3)

C·x power (k=13) = -42

term[13] := mul(-42, -8192)

term[13] = 344064

choose 14 positions out of 14

C(14,14)=87178291200/(87178291200·1)=1

x^0: (-3)^0

x^0 = 1

y^14: (-2)^14

y^14 = 16384

C·x power (k=14) := mul(1, 1)

C·x power (k=14) = 1

term[14] := mul(1, 16384)

term[14] = 16384

expanded sum := sum(4782969, 44641044, 193444524, 515852064, 945728784, 1260971712, 1260971712, 960740352, 560431872, 249080832, 83026944, 20127744, 3354624, 344064, 16384)

expanded sum = 6103515625

direct=6103515625; expanded=6103515625

identity holds: Yes

Reverse checks

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

(14, -3, -2, 6103515625, 6103515625, 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.