Question 98: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=22, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=22, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

Definition and scope

n is a nonnegative integer and x,y commute. For n=0 use the polynomial empty-product identity.

• mko-binomial-theorem

Step-by-step derivation

n=22≥0; x,y commute

(x+y)^22 is a finite product of 22 factors

choose exactly 0 of 22 factors to contribute y

C(22,0)=1124000727777607680000/(1·1124000727777607680000)=1

C(22,0)=1; choose y 0 times

1·x^22·y^0

choose exactly 1 of 22 factors to contribute y

C(22,1)=1124000727777607680000/(1·51090942171709440000)=22

C(22,1)=22; choose y 1 times

22·x^21·y^1

choose exactly 2 of 22 factors to contribute y

C(22,2)=1124000727777607680000/(2·2432902008176640000)=231

C(22,2)=231; choose y 2 times

231·x^20·y^2

choose exactly 3 of 22 factors to contribute y

C(22,3)=1124000727777607680000/(6·121645100408832000)=1540

C(22,3)=1540; choose y 3 times

1540·x^19·y^3

choose exactly 4 of 22 factors to contribute y

C(22,4)=1124000727777607680000/(24·6402373705728000)=7315

C(22,4)=7315; choose y 4 times

7315·x^18·y^4

choose exactly 5 of 22 factors to contribute y

C(22,5)=1124000727777607680000/(120·355687428096000)=26334

C(22,5)=26334; choose y 5 times

26334·x^17·y^5

choose exactly 6 of 22 factors to contribute y

C(22,6)=1124000727777607680000/(720·20922789888000)=74613

C(22,6)=74613; choose y 6 times

74613·x^16·y^6

choose exactly 7 of 22 factors to contribute y

C(22,7)=1124000727777607680000/(5040·1307674368000)=170544

C(22,7)=170544; choose y 7 times

170544·x^15·y^7

choose exactly 8 of 22 factors to contribute y

C(22,8)=1124000727777607680000/(40320·87178291200)=319770

C(22,8)=319770; choose y 8 times

319770·x^14·y^8

choose exactly 9 of 22 factors to contribute y

C(22,9)=1124000727777607680000/(362880·6227020800)=497420

C(22,9)=497420; choose y 9 times

497420·x^13·y^9

choose exactly 10 of 22 factors to contribute y

C(22,10)=1124000727777607680000/(3628800·479001600)=646646

C(22,10)=646646; choose y 10 times

646646·x^12·y^10

choose exactly 11 of 22 factors to contribute y

C(22,11)=1124000727777607680000/(39916800·39916800)=705432

C(22,11)=705432; choose y 11 times

705432·x^11·y^11

choose exactly 12 of 22 factors to contribute y

C(22,12)=1124000727777607680000/(479001600·3628800)=646646

C(22,12)=646646; choose y 12 times

646646·x^10·y^12

choose exactly 13 of 22 factors to contribute y

C(22,13)=1124000727777607680000/(6227020800·362880)=497420

C(22,13)=497420; choose y 13 times

497420·x^9·y^13

choose exactly 14 of 22 factors to contribute y

C(22,14)=1124000727777607680000/(87178291200·40320)=319770

C(22,14)=319770; choose y 14 times

319770·x^8·y^14

choose exactly 15 of 22 factors to contribute y

C(22,15)=1124000727777607680000/(1307674368000·5040)=170544

C(22,15)=170544; choose y 15 times

170544·x^7·y^15

choose exactly 16 of 22 factors to contribute y

C(22,16)=1124000727777607680000/(20922789888000·720)=74613

C(22,16)=74613; choose y 16 times

74613·x^6·y^16

choose exactly 17 of 22 factors to contribute y

C(22,17)=1124000727777607680000/(355687428096000·120)=26334

C(22,17)=26334; choose y 17 times

26334·x^5·y^17

choose exactly 18 of 22 factors to contribute y

C(22,18)=1124000727777607680000/(6402373705728000·24)=7315

C(22,18)=7315; choose y 18 times

7315·x^4·y^18

choose exactly 19 of 22 factors to contribute y

C(22,19)=1124000727777607680000/(121645100408832000·6)=1540

C(22,19)=1540; choose y 19 times

1540·x^3·y^19

choose exactly 20 of 22 factors to contribute y

C(22,20)=1124000727777607680000/(2432902008176640000·2)=231

C(22,20)=231; choose y 20 times

231·x^2·y^20

choose exactly 21 of 22 factors to contribute y

C(22,21)=1124000727777607680000/(51090942171709440000·1)=22

C(22,21)=22; choose y 21 times

22·x^1·y^21

choose exactly 22 of 22 factors to contribute y

C(22,22)=1124000727777607680000/(1124000727777607680000·1)=1

C(22,22)=1; choose y 22 times

1·x^0·y^22

all k=0 through 22 have been accounted for

(x+y)^22=1·x^22·y^0 + 22·x^21·y^1 + 231·x^20·y^2 + 1540·x^19·y^3 + 7315·x^18·y^4 + 26334·x^17·y^5 + 74613·x^16·y^6 + 170544·x^15·y^7 + 319770·x^14·y^8 + 497420·x^13·y^9 + 646646·x^12·y^10 + 705432·x^11·y^11 + 646646·x^10·y^12 + 497420·x^9·y^13 + 319770·x^8·y^14 + 170544·x^7·y^15 + 74613·x^6·y^16 + 26334·x^5·y^17 + 7315·x^4·y^18 + 1540·x^3·y^19 + 231·x^2·y^20 + 22·x^1·y^21 + 1·x^0·y^22

Reverse checks

• S048 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• S047 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S046 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S045 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S044 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S043 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S042 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S041 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S040 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S039 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S038 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S037 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S036 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S035 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S034 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S033 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S032 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S031 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S030 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S029 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S028 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S027 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S026 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S025 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S024 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S023 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S022 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S021 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S020 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S019 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S018 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S017 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S016 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S015 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S014 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S013 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S012 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S011 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S010 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S009 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S008 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S007 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S006 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S005 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S004 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S003 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S002 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S001 · premise_and_result_audit — This is not an infinite expansion for a noninteger power or a claim for noncommuting multiplication.

Result

(22, [[0, 1, 22, 0], [1, 22, 21, 1], [2, 231, 20, 2], [3, 1540, 19, 3], [4, 7315, 18, 4], [5, 26334, 17, 5], [6, 74613, 16, 6], [7, 170544, 15, 7], [8, 319770, 14, 8], [9, 497420, 13, 9], [10, 646646, 12, 10], [11, 705432, 11, 11], [12, 646646, 10, 12], [13, 497420, 9, 13], [14, 319770, 8, 14], [15, 170544, 7, 15], [16, 74613, 6, 16], [17, 26334, 5, 17], [18, 7315, 4, 18], [19, 1540, 3, 19], [20, 231, 2, 20], [21, 22, 1, 21], [22, 1, 0, 22]], complete_k_zero_through_n_coefficients)

The full expansion consists of all displayed monomials, in the same order as the answer rows.

This question review and source binding

Review date: 2026-10-24

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.