Question 96: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=20, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=20, 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=20≥0; x,y commute

(x+y)^20 is a finite product of 20 factors

choose exactly 0 of 20 factors to contribute y

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

C(20,0)=1; choose y 0 times

1·x^20·y^0

choose exactly 1 of 20 factors to contribute y

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

C(20,1)=20; choose y 1 times

20·x^19·y^1

choose exactly 2 of 20 factors to contribute y

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

C(20,2)=190; choose y 2 times

190·x^18·y^2

choose exactly 3 of 20 factors to contribute y

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

C(20,3)=1140; choose y 3 times

1140·x^17·y^3

choose exactly 4 of 20 factors to contribute y

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

C(20,4)=4845; choose y 4 times

4845·x^16·y^4

choose exactly 5 of 20 factors to contribute y

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

C(20,5)=15504; choose y 5 times

15504·x^15·y^5

choose exactly 6 of 20 factors to contribute y

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

C(20,6)=38760; choose y 6 times

38760·x^14·y^6

choose exactly 7 of 20 factors to contribute y

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

C(20,7)=77520; choose y 7 times

77520·x^13·y^7

choose exactly 8 of 20 factors to contribute y

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

C(20,8)=125970; choose y 8 times

125970·x^12·y^8

choose exactly 9 of 20 factors to contribute y

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

C(20,9)=167960; choose y 9 times

167960·x^11·y^9

choose exactly 10 of 20 factors to contribute y

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

C(20,10)=184756; choose y 10 times

184756·x^10·y^10

choose exactly 11 of 20 factors to contribute y

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

C(20,11)=167960; choose y 11 times

167960·x^9·y^11

choose exactly 12 of 20 factors to contribute y

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

C(20,12)=125970; choose y 12 times

125970·x^8·y^12

choose exactly 13 of 20 factors to contribute y

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

C(20,13)=77520; choose y 13 times

77520·x^7·y^13

choose exactly 14 of 20 factors to contribute y

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

C(20,14)=38760; choose y 14 times

38760·x^6·y^14

choose exactly 15 of 20 factors to contribute y

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

C(20,15)=15504; choose y 15 times

15504·x^5·y^15

choose exactly 16 of 20 factors to contribute y

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

C(20,16)=4845; choose y 16 times

4845·x^4·y^16

choose exactly 17 of 20 factors to contribute y

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

C(20,17)=1140; choose y 17 times

1140·x^3·y^17

choose exactly 18 of 20 factors to contribute y

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

C(20,18)=190; choose y 18 times

190·x^2·y^18

choose exactly 19 of 20 factors to contribute y

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

C(20,19)=20; choose y 19 times

20·x^1·y^19

choose exactly 20 of 20 factors to contribute y

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

C(20,20)=1; choose y 20 times

1·x^0·y^20

all k=0 through 20 have been accounted for

(x+y)^20=1·x^20·y^0 + 20·x^19·y^1 + 190·x^18·y^2 + 1140·x^17·y^3 + 4845·x^16·y^4 + 15504·x^15·y^5 + 38760·x^14·y^6 + 77520·x^13·y^7 + 125970·x^12·y^8 + 167960·x^11·y^9 + 184756·x^10·y^10 + 167960·x^9·y^11 + 125970·x^8·y^12 + 77520·x^7·y^13 + 38760·x^6·y^14 + 15504·x^5·y^15 + 4845·x^4·y^16 + 1140·x^3·y^17 + 190·x^2·y^18 + 20·x^1·y^19 + 1·x^0·y^20

Reverse checks

• S044 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• 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

(20, [[0, 1, 20, 0], [1, 20, 19, 1], [2, 190, 18, 2], [3, 1140, 17, 3], [4, 4845, 16, 4], [5, 15504, 15, 5], [6, 38760, 14, 6], [7, 77520, 13, 7], [8, 125970, 12, 8], [9, 167960, 11, 9], [10, 184756, 10, 10], [11, 167960, 9, 11], [12, 125970, 8, 12], [13, 77520, 7, 13], [14, 38760, 6, 14], [15, 15504, 5, 15], [16, 4845, 4, 16], [17, 1140, 3, 17], [18, 190, 2, 18], [19, 20, 1, 19], [20, 1, 0, 20]], 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.