Question 446: Compare the Whole Binomial Sum with an Exact Integer Power

Original question and givens

Use natural exponent n=20, exact commuting integers x=3, y=2. Give (n, x, y, power, expansion_sum, equal).

• G001 — Use natural exponent n=20, 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=20≥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)^20

(x+y)^n = 95367431640625

choose 0 positions out of 20

C(20,0)=2432902008176640000/(1·2432902008176640000)=1

x^20: (3)^20

x^20 = 3486784401

y^0: (2)^0

y^0 = 1

C·x power (k=0) := mul(1, 3486784401)

C·x power (k=0) = 3486784401

term[0] := mul(3486784401, 1)

term[0] = 3486784401

choose 1 positions out of 20

C(20,1)=2432902008176640000/(1·121645100408832000)=20

x^19: (3)^19

x^19 = 1162261467

y^1: (2)^1

y^1 = 2

C·x power (k=1) := mul(20, 1162261467)

C·x power (k=1) = 23245229340

term[1] := mul(23245229340, 2)

term[1] = 46490458680

choose 2 positions out of 20

C(20,2)=2432902008176640000/(2·6402373705728000)=190

x^18: (3)^18

x^18 = 387420489

y^2: (2)^2

y^2 = 4

C·x power (k=2) := mul(190, 387420489)

C·x power (k=2) = 73609892910

term[2] := mul(73609892910, 4)

term[2] = 294439571640

choose 3 positions out of 20

C(20,3)=2432902008176640000/(6·355687428096000)=1140

x^17: (3)^17

x^17 = 129140163

y^3: (2)^3

y^3 = 8

C·x power (k=3) := mul(1140, 129140163)

C·x power (k=3) = 147219785820

term[3] := mul(147219785820, 8)

term[3] = 1177758286560

choose 4 positions out of 20

C(20,4)=2432902008176640000/(24·20922789888000)=4845

x^16: (3)^16

x^16 = 43046721

y^4: (2)^4

y^4 = 16

C·x power (k=4) := mul(4845, 43046721)

C·x power (k=4) = 208561363245

term[4] := mul(208561363245, 16)

term[4] = 3336981811920

choose 5 positions out of 20

C(20,5)=2432902008176640000/(120·1307674368000)=15504

x^15: (3)^15

x^15 = 14348907

y^5: (2)^5

y^5 = 32

C·x power (k=5) := mul(15504, 14348907)

C·x power (k=5) = 222465454128

term[5] := mul(222465454128, 32)

term[5] = 7118894532096

choose 6 positions out of 20

C(20,6)=2432902008176640000/(720·87178291200)=38760

x^14: (3)^14

x^14 = 4782969

y^6: (2)^6

y^6 = 64

C·x power (k=6) := mul(38760, 4782969)

C·x power (k=6) = 185387878440

term[6] := mul(185387878440, 64)

term[6] = 11864824220160

choose 7 positions out of 20

C(20,7)=2432902008176640000/(5040·6227020800)=77520

x^13: (3)^13

x^13 = 1594323

y^7: (2)^7

y^7 = 128

C·x power (k=7) := mul(77520, 1594323)

C·x power (k=7) = 123591918960

term[7] := mul(123591918960, 128)

term[7] = 15819765626880

choose 8 positions out of 20

C(20,8)=2432902008176640000/(40320·479001600)=125970

x^12: (3)^12

x^12 = 531441

y^8: (2)^8

y^8 = 256

C·x power (k=8) := mul(125970, 531441)

C·x power (k=8) = 66945622770

term[8] := mul(66945622770, 256)

term[8] = 17138079429120

choose 9 positions out of 20

C(20,9)=2432902008176640000/(362880·39916800)=167960

x^11: (3)^11

x^11 = 177147

y^9: (2)^9

y^9 = 512

C·x power (k=9) := mul(167960, 177147)

C·x power (k=9) = 29753610120

term[9] := mul(29753610120, 512)

term[9] = 15233848381440

choose 10 positions out of 20

C(20,10)=2432902008176640000/(3628800·3628800)=184756

x^10: (3)^10

x^10 = 59049

y^10: (2)^10

y^10 = 1024

C·x power (k=10) := mul(184756, 59049)

C·x power (k=10) = 10909657044

term[10] := mul(10909657044, 1024)

term[10] = 11171488813056

choose 11 positions out of 20

C(20,11)=2432902008176640000/(39916800·362880)=167960

x^9: (3)^9

x^9 = 19683

y^11: (2)^11

y^11 = 2048

C·x power (k=11) := mul(167960, 19683)

C·x power (k=11) = 3305956680

term[11] := mul(3305956680, 2048)

term[11] = 6770599280640

choose 12 positions out of 20

C(20,12)=2432902008176640000/(479001600·40320)=125970

x^8: (3)^8

x^8 = 6561

y^12: (2)^12

y^12 = 4096

C·x power (k=12) := mul(125970, 6561)

C·x power (k=12) = 826489170

term[12] := mul(826489170, 4096)

term[12] = 3385299640320

choose 13 positions out of 20

C(20,13)=2432902008176640000/(6227020800·5040)=77520

x^7: (3)^7

x^7 = 2187

y^13: (2)^13

y^13 = 8192

C·x power (k=13) := mul(77520, 2187)

C·x power (k=13) = 169536240

term[13] := mul(169536240, 8192)

term[13] = 1388840878080

choose 14 positions out of 20

C(20,14)=2432902008176640000/(87178291200·720)=38760

x^6: (3)^6

x^6 = 729

y^14: (2)^14

y^14 = 16384

C·x power (k=14) := mul(38760, 729)

C·x power (k=14) = 28256040

term[14] := mul(28256040, 16384)

term[14] = 462946959360

choose 15 positions out of 20

C(20,15)=2432902008176640000/(1307674368000·120)=15504

x^5: (3)^5

x^5 = 243

y^15: (2)^15

y^15 = 32768

C·x power (k=15) := mul(15504, 243)

C·x power (k=15) = 3767472

term[15] := mul(3767472, 32768)

term[15] = 123452522496

choose 16 positions out of 20

C(20,16)=2432902008176640000/(20922789888000·24)=4845

x^4: (3)^4

x^4 = 81

y^16: (2)^16

y^16 = 65536

C·x power (k=16) := mul(4845, 81)

C·x power (k=16) = 392445

term[16] := mul(392445, 65536)

term[16] = 25719275520

choose 17 positions out of 20

C(20,17)=2432902008176640000/(355687428096000·6)=1140

x^3: (3)^3

x^3 = 27

y^17: (2)^17

y^17 = 131072

C·x power (k=17) := mul(1140, 27)

C·x power (k=17) = 30780

term[17] := mul(30780, 131072)

term[17] = 4034396160

choose 18 positions out of 20

C(20,18)=2432902008176640000/(6402373705728000·2)=190

x^2: (3)^2

x^2 = 9

y^18: (2)^18

y^18 = 262144

C·x power (k=18) := mul(190, 9)

C·x power (k=18) = 1710

term[18] := mul(1710, 262144)

term[18] = 448266240

choose 19 positions out of 20

C(20,19)=2432902008176640000/(121645100408832000·1)=20

x^1: (3)^1

x^1 = 3

y^19: (2)^19

y^19 = 524288

C·x power (k=19) := mul(20, 3)

C·x power (k=19) = 60

term[19] := mul(60, 524288)

term[19] = 31457280

choose 20 positions out of 20

C(20,20)=2432902008176640000/(2432902008176640000·1)=1

x^0: (3)^0

x^0 = 1

y^20: (2)^20

y^20 = 1048576

C·x power (k=20) := mul(1, 1)

C·x power (k=20) = 1

term[20] := mul(1, 1048576)

term[20] = 1048576

expanded sum := sum(3486784401, 46490458680, 294439571640, 1177758286560, 3336981811920, 7118894532096, 11864824220160, 15819765626880, 17138079429120, 15233848381440, 11171488813056, 6770599280640, 3385299640320, 1388840878080, 462946959360, 123452522496, 25719275520, 4034396160, 448266240, 31457280, 1048576)

expanded sum = 95367431640625

direct=95367431640625; expanded=95367431640625

identity holds: Yes

Reverse checks

• S110 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S109 · premise_and_result_audit — Recompute expanded sum with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute term[20] with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute C·x power (k=20) with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S105 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S104 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S103 · premise_and_result_audit — Recompute term[19] with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute C·x power (k=19) with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• 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 — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• 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

(20, 3, 2, 95367431640625, 95367431640625, 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.