Question 95: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=19, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=19, 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=19≥0; x,y commute

(x+y)^19 is a finite product of 19 factors

choose exactly 0 of 19 factors to contribute y

C(19,0)=121645100408832000/(1·121645100408832000)=1

C(19,0)=1; choose y 0 times

1·x^19·y^0

choose exactly 1 of 19 factors to contribute y

C(19,1)=121645100408832000/(1·6402373705728000)=19

C(19,1)=19; choose y 1 times

19·x^18·y^1

choose exactly 2 of 19 factors to contribute y

C(19,2)=121645100408832000/(2·355687428096000)=171

C(19,2)=171; choose y 2 times

171·x^17·y^2

choose exactly 3 of 19 factors to contribute y

C(19,3)=121645100408832000/(6·20922789888000)=969

C(19,3)=969; choose y 3 times

969·x^16·y^3

choose exactly 4 of 19 factors to contribute y

C(19,4)=121645100408832000/(24·1307674368000)=3876

C(19,4)=3876; choose y 4 times

3876·x^15·y^4

choose exactly 5 of 19 factors to contribute y

C(19,5)=121645100408832000/(120·87178291200)=11628

C(19,5)=11628; choose y 5 times

11628·x^14·y^5

choose exactly 6 of 19 factors to contribute y

C(19,6)=121645100408832000/(720·6227020800)=27132

C(19,6)=27132; choose y 6 times

27132·x^13·y^6

choose exactly 7 of 19 factors to contribute y

C(19,7)=121645100408832000/(5040·479001600)=50388

C(19,7)=50388; choose y 7 times

50388·x^12·y^7

choose exactly 8 of 19 factors to contribute y

C(19,8)=121645100408832000/(40320·39916800)=75582

C(19,8)=75582; choose y 8 times

75582·x^11·y^8

choose exactly 9 of 19 factors to contribute y

C(19,9)=121645100408832000/(362880·3628800)=92378

C(19,9)=92378; choose y 9 times

92378·x^10·y^9

choose exactly 10 of 19 factors to contribute y

C(19,10)=121645100408832000/(3628800·362880)=92378

C(19,10)=92378; choose y 10 times

92378·x^9·y^10

choose exactly 11 of 19 factors to contribute y

C(19,11)=121645100408832000/(39916800·40320)=75582

C(19,11)=75582; choose y 11 times

75582·x^8·y^11

choose exactly 12 of 19 factors to contribute y

C(19,12)=121645100408832000/(479001600·5040)=50388

C(19,12)=50388; choose y 12 times

50388·x^7·y^12

choose exactly 13 of 19 factors to contribute y

C(19,13)=121645100408832000/(6227020800·720)=27132

C(19,13)=27132; choose y 13 times

27132·x^6·y^13

choose exactly 14 of 19 factors to contribute y

C(19,14)=121645100408832000/(87178291200·120)=11628

C(19,14)=11628; choose y 14 times

11628·x^5·y^14

choose exactly 15 of 19 factors to contribute y

C(19,15)=121645100408832000/(1307674368000·24)=3876

C(19,15)=3876; choose y 15 times

3876·x^4·y^15

choose exactly 16 of 19 factors to contribute y

C(19,16)=121645100408832000/(20922789888000·6)=969

C(19,16)=969; choose y 16 times

969·x^3·y^16

choose exactly 17 of 19 factors to contribute y

C(19,17)=121645100408832000/(355687428096000·2)=171

C(19,17)=171; choose y 17 times

171·x^2·y^17

choose exactly 18 of 19 factors to contribute y

C(19,18)=121645100408832000/(6402373705728000·1)=19

C(19,18)=19; choose y 18 times

19·x^1·y^18

choose exactly 19 of 19 factors to contribute y

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

C(19,19)=1; choose y 19 times

1·x^0·y^19

all k=0 through 19 have been accounted for

(x+y)^19=1·x^19·y^0 + 19·x^18·y^1 + 171·x^17·y^2 + 969·x^16·y^3 + 3876·x^15·y^4 + 11628·x^14·y^5 + 27132·x^13·y^6 + 50388·x^12·y^7 + 75582·x^11·y^8 + 92378·x^10·y^9 + 92378·x^9·y^10 + 75582·x^8·y^11 + 50388·x^7·y^12 + 27132·x^6·y^13 + 11628·x^5·y^14 + 3876·x^4·y^15 + 969·x^3·y^16 + 171·x^2·y^17 + 19·x^1·y^18 + 1·x^0·y^19

Reverse checks

• S042 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• 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

(19, [[0, 1, 19, 0], [1, 19, 18, 1], [2, 171, 17, 2], [3, 969, 16, 3], [4, 3876, 15, 4], [5, 11628, 14, 5], [6, 27132, 13, 6], [7, 50388, 12, 7], [8, 75582, 11, 8], [9, 92378, 10, 9], [10, 92378, 9, 10], [11, 75582, 8, 11], [12, 50388, 7, 12], [13, 27132, 6, 13], [14, 11628, 5, 14], [15, 3876, 4, 15], [16, 969, 3, 16], [17, 171, 2, 17], [18, 19, 1, 18], [19, 1, 0, 19]], 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.