Question 444: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=18, exact commuting integers x=1, y=-1. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=18, 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=18≥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)^18

(x+y)^n = 0

choose 0 positions out of 18

C(18,0)=6402373705728000/(1·6402373705728000)=1

x^18: (1)^18

x^18 = 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 18

C(18,1)=6402373705728000/(1·355687428096000)=18

x^17: (1)^17

x^17 = 1

y^1: (-1)^1

y^1 = -1

C·x power (k=1) := mul(18, 1)

C·x power (k=1) = 18

term[1] := mul(18, -1)

term[1] = -18

choose 2 positions out of 18

C(18,2)=6402373705728000/(2·20922789888000)=153

x^16: (1)^16

x^16 = 1

y^2: (-1)^2

y^2 = 1

C·x power (k=2) := mul(153, 1)

C·x power (k=2) = 153

term[2] := mul(153, 1)

term[2] = 153

choose 3 positions out of 18

C(18,3)=6402373705728000/(6·1307674368000)=816

x^15: (1)^15

x^15 = 1

y^3: (-1)^3

y^3 = -1

C·x power (k=3) := mul(816, 1)

C·x power (k=3) = 816

term[3] := mul(816, -1)

term[3] = -816

choose 4 positions out of 18

C(18,4)=6402373705728000/(24·87178291200)=3060

x^14: (1)^14

x^14 = 1

y^4: (-1)^4

y^4 = 1

C·x power (k=4) := mul(3060, 1)

C·x power (k=4) = 3060

term[4] := mul(3060, 1)

term[4] = 3060

choose 5 positions out of 18

C(18,5)=6402373705728000/(120·6227020800)=8568

x^13: (1)^13

x^13 = 1

y^5: (-1)^5

y^5 = -1

C·x power (k=5) := mul(8568, 1)

C·x power (k=5) = 8568

term[5] := mul(8568, -1)

term[5] = -8568

choose 6 positions out of 18

C(18,6)=6402373705728000/(720·479001600)=18564

x^12: (1)^12

x^12 = 1

y^6: (-1)^6

y^6 = 1

C·x power (k=6) := mul(18564, 1)

C·x power (k=6) = 18564

term[6] := mul(18564, 1)

term[6] = 18564

choose 7 positions out of 18

C(18,7)=6402373705728000/(5040·39916800)=31824

x^11: (1)^11

x^11 = 1

y^7: (-1)^7

y^7 = -1

C·x power (k=7) := mul(31824, 1)

C·x power (k=7) = 31824

term[7] := mul(31824, -1)

term[7] = -31824

choose 8 positions out of 18

C(18,8)=6402373705728000/(40320·3628800)=43758

x^10: (1)^10

x^10 = 1

y^8: (-1)^8

y^8 = 1

C·x power (k=8) := mul(43758, 1)

C·x power (k=8) = 43758

term[8] := mul(43758, 1)

term[8] = 43758

choose 9 positions out of 18

C(18,9)=6402373705728000/(362880·362880)=48620

x^9: (1)^9

x^9 = 1

y^9: (-1)^9

y^9 = -1

C·x power (k=9) := mul(48620, 1)

C·x power (k=9) = 48620

term[9] := mul(48620, -1)

term[9] = -48620

choose 10 positions out of 18

C(18,10)=6402373705728000/(3628800·40320)=43758

x^8: (1)^8

x^8 = 1

y^10: (-1)^10

y^10 = 1

C·x power (k=10) := mul(43758, 1)

C·x power (k=10) = 43758

term[10] := mul(43758, 1)

term[10] = 43758

choose 11 positions out of 18

C(18,11)=6402373705728000/(39916800·5040)=31824

x^7: (1)^7

x^7 = 1

y^11: (-1)^11

y^11 = -1

C·x power (k=11) := mul(31824, 1)

C·x power (k=11) = 31824

term[11] := mul(31824, -1)

term[11] = -31824

choose 12 positions out of 18

C(18,12)=6402373705728000/(479001600·720)=18564

x^6: (1)^6

x^6 = 1

y^12: (-1)^12

y^12 = 1

C·x power (k=12) := mul(18564, 1)

C·x power (k=12) = 18564

term[12] := mul(18564, 1)

term[12] = 18564

choose 13 positions out of 18

C(18,13)=6402373705728000/(6227020800·120)=8568

x^5: (1)^5

x^5 = 1

y^13: (-1)^13

y^13 = -1

C·x power (k=13) := mul(8568, 1)

C·x power (k=13) = 8568

term[13] := mul(8568, -1)

term[13] = -8568

choose 14 positions out of 18

C(18,14)=6402373705728000/(87178291200·24)=3060

x^4: (1)^4

x^4 = 1

y^14: (-1)^14

y^14 = 1

C·x power (k=14) := mul(3060, 1)

C·x power (k=14) = 3060

term[14] := mul(3060, 1)

term[14] = 3060

choose 15 positions out of 18

C(18,15)=6402373705728000/(1307674368000·6)=816

x^3: (1)^3

x^3 = 1

y^15: (-1)^15

y^15 = -1

C·x power (k=15) := mul(816, 1)

C·x power (k=15) = 816

term[15] := mul(816, -1)

term[15] = -816

choose 16 positions out of 18

C(18,16)=6402373705728000/(20922789888000·2)=153

x^2: (1)^2

x^2 = 1

y^16: (-1)^16

y^16 = 1

C·x power (k=16) := mul(153, 1)

C·x power (k=16) = 153

term[16] := mul(153, 1)

term[16] = 153

choose 17 positions out of 18

C(18,17)=6402373705728000/(355687428096000·1)=18

x^1: (1)^1

x^1 = 1

y^17: (-1)^17

y^17 = -1

C·x power (k=17) := mul(18, 1)

C·x power (k=17) = 18

term[17] := mul(18, -1)

term[17] = -18

choose 18 positions out of 18

C(18,18)=6402373705728000/(6402373705728000·1)=1

x^0: (1)^0

x^0 = 1

y^18: (-1)^18

y^18 = 1

C·x power (k=18) := mul(1, 1)

C·x power (k=18) = 1

term[18] := mul(1, 1)

term[18] = 1

expanded sum := sum(1, -18, 153, -816, 3060, -8568, 18564, -31824, 43758, -48620, 43758, -31824, 18564, -8568, 3060, -816, 153, -18, 1)

expanded sum = 0

direct=0; expanded=0

identity holds: Yes

Reverse checks

• S100 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S099 · premise_and_result_audit — Recompute expanded sum with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute term[18] with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute C·x power (k=18) with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S095 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S094 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S093 · premise_and_result_audit — Recompute term[17] with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute C·x power (k=17) with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• 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 — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• 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

(18, 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.