Question 97: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=21, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=21, 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=21≥0; x,y commute

(x+y)^21 is a finite product of 21 factors

choose exactly 0 of 21 factors to contribute y

C(21,0)=51090942171709440000/(1·51090942171709440000)=1

C(21,0)=1; choose y 0 times

1·x^21·y^0

choose exactly 1 of 21 factors to contribute y

C(21,1)=51090942171709440000/(1·2432902008176640000)=21

C(21,1)=21; choose y 1 times

21·x^20·y^1

choose exactly 2 of 21 factors to contribute y

C(21,2)=51090942171709440000/(2·121645100408832000)=210

C(21,2)=210; choose y 2 times

210·x^19·y^2

choose exactly 3 of 21 factors to contribute y

C(21,3)=51090942171709440000/(6·6402373705728000)=1330

C(21,3)=1330; choose y 3 times

1330·x^18·y^3

choose exactly 4 of 21 factors to contribute y

C(21,4)=51090942171709440000/(24·355687428096000)=5985

C(21,4)=5985; choose y 4 times

5985·x^17·y^4

choose exactly 5 of 21 factors to contribute y

C(21,5)=51090942171709440000/(120·20922789888000)=20349

C(21,5)=20349; choose y 5 times

20349·x^16·y^5

choose exactly 6 of 21 factors to contribute y

C(21,6)=51090942171709440000/(720·1307674368000)=54264

C(21,6)=54264; choose y 6 times

54264·x^15·y^6

choose exactly 7 of 21 factors to contribute y

C(21,7)=51090942171709440000/(5040·87178291200)=116280

C(21,7)=116280; choose y 7 times

116280·x^14·y^7

choose exactly 8 of 21 factors to contribute y

C(21,8)=51090942171709440000/(40320·6227020800)=203490

C(21,8)=203490; choose y 8 times

203490·x^13·y^8

choose exactly 9 of 21 factors to contribute y

C(21,9)=51090942171709440000/(362880·479001600)=293930

C(21,9)=293930; choose y 9 times

293930·x^12·y^9

choose exactly 10 of 21 factors to contribute y

C(21,10)=51090942171709440000/(3628800·39916800)=352716

C(21,10)=352716; choose y 10 times

352716·x^11·y^10

choose exactly 11 of 21 factors to contribute y

C(21,11)=51090942171709440000/(39916800·3628800)=352716

C(21,11)=352716; choose y 11 times

352716·x^10·y^11

choose exactly 12 of 21 factors to contribute y

C(21,12)=51090942171709440000/(479001600·362880)=293930

C(21,12)=293930; choose y 12 times

293930·x^9·y^12

choose exactly 13 of 21 factors to contribute y

C(21,13)=51090942171709440000/(6227020800·40320)=203490

C(21,13)=203490; choose y 13 times

203490·x^8·y^13

choose exactly 14 of 21 factors to contribute y

C(21,14)=51090942171709440000/(87178291200·5040)=116280

C(21,14)=116280; choose y 14 times

116280·x^7·y^14

choose exactly 15 of 21 factors to contribute y

C(21,15)=51090942171709440000/(1307674368000·720)=54264

C(21,15)=54264; choose y 15 times

54264·x^6·y^15

choose exactly 16 of 21 factors to contribute y

C(21,16)=51090942171709440000/(20922789888000·120)=20349

C(21,16)=20349; choose y 16 times

20349·x^5·y^16

choose exactly 17 of 21 factors to contribute y

C(21,17)=51090942171709440000/(355687428096000·24)=5985

C(21,17)=5985; choose y 17 times

5985·x^4·y^17

choose exactly 18 of 21 factors to contribute y

C(21,18)=51090942171709440000/(6402373705728000·6)=1330

C(21,18)=1330; choose y 18 times

1330·x^3·y^18

choose exactly 19 of 21 factors to contribute y

C(21,19)=51090942171709440000/(121645100408832000·2)=210

C(21,19)=210; choose y 19 times

210·x^2·y^19

choose exactly 20 of 21 factors to contribute y

C(21,20)=51090942171709440000/(2432902008176640000·1)=21

C(21,20)=21; choose y 20 times

21·x^1·y^20

choose exactly 21 of 21 factors to contribute y

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

C(21,21)=1; choose y 21 times

1·x^0·y^21

all k=0 through 21 have been accounted for

(x+y)^21=1·x^21·y^0 + 21·x^20·y^1 + 210·x^19·y^2 + 1330·x^18·y^3 + 5985·x^17·y^4 + 20349·x^16·y^5 + 54264·x^15·y^6 + 116280·x^14·y^7 + 203490·x^13·y^8 + 293930·x^12·y^9 + 352716·x^11·y^10 + 352716·x^10·y^11 + 293930·x^9·y^12 + 203490·x^8·y^13 + 116280·x^7·y^14 + 54264·x^6·y^15 + 20349·x^5·y^16 + 5985·x^4·y^17 + 1330·x^3·y^18 + 210·x^2·y^19 + 21·x^1·y^20 + 1·x^0·y^21

Reverse checks

• S046 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• 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

(21, [[0, 1, 21, 0], [1, 21, 20, 1], [2, 210, 19, 2], [3, 1330, 18, 3], [4, 5985, 17, 4], [5, 20349, 16, 5], [6, 54264, 15, 6], [7, 116280, 14, 7], [8, 203490, 13, 8], [9, 293930, 12, 9], [10, 352716, 11, 10], [11, 352716, 10, 11], [12, 293930, 9, 12], [13, 203490, 8, 13], [14, 116280, 7, 14], [15, 54264, 6, 15], [16, 20349, 5, 16], [17, 5985, 4, 17], [18, 1330, 3, 18], [19, 210, 2, 19], [20, 21, 1, 20], [21, 1, 0, 21]], 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.