Question 94: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=18, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=18, 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=18≥0; x,y commute

(x+y)^18 is a finite product of 18 factors

choose exactly 0 of 18 factors to contribute y

C(18,0)=6402373705728000/(1·6402373705728000)=1

C(18,0)=1; choose y 0 times

1·x^18·y^0

choose exactly 1 of 18 factors to contribute y

C(18,1)=6402373705728000/(1·355687428096000)=18

C(18,1)=18; choose y 1 times

18·x^17·y^1

choose exactly 2 of 18 factors to contribute y

C(18,2)=6402373705728000/(2·20922789888000)=153

C(18,2)=153; choose y 2 times

153·x^16·y^2

choose exactly 3 of 18 factors to contribute y

C(18,3)=6402373705728000/(6·1307674368000)=816

C(18,3)=816; choose y 3 times

816·x^15·y^3

choose exactly 4 of 18 factors to contribute y

C(18,4)=6402373705728000/(24·87178291200)=3060

C(18,4)=3060; choose y 4 times

3060·x^14·y^4

choose exactly 5 of 18 factors to contribute y

C(18,5)=6402373705728000/(120·6227020800)=8568

C(18,5)=8568; choose y 5 times

8568·x^13·y^5

choose exactly 6 of 18 factors to contribute y

C(18,6)=6402373705728000/(720·479001600)=18564

C(18,6)=18564; choose y 6 times

18564·x^12·y^6

choose exactly 7 of 18 factors to contribute y

C(18,7)=6402373705728000/(5040·39916800)=31824

C(18,7)=31824; choose y 7 times

31824·x^11·y^7

choose exactly 8 of 18 factors to contribute y

C(18,8)=6402373705728000/(40320·3628800)=43758

C(18,8)=43758; choose y 8 times

43758·x^10·y^8

choose exactly 9 of 18 factors to contribute y

C(18,9)=6402373705728000/(362880·362880)=48620

C(18,9)=48620; choose y 9 times

48620·x^9·y^9

choose exactly 10 of 18 factors to contribute y

C(18,10)=6402373705728000/(3628800·40320)=43758

C(18,10)=43758; choose y 10 times

43758·x^8·y^10

choose exactly 11 of 18 factors to contribute y

C(18,11)=6402373705728000/(39916800·5040)=31824

C(18,11)=31824; choose y 11 times

31824·x^7·y^11

choose exactly 12 of 18 factors to contribute y

C(18,12)=6402373705728000/(479001600·720)=18564

C(18,12)=18564; choose y 12 times

18564·x^6·y^12

choose exactly 13 of 18 factors to contribute y

C(18,13)=6402373705728000/(6227020800·120)=8568

C(18,13)=8568; choose y 13 times

8568·x^5·y^13

choose exactly 14 of 18 factors to contribute y

C(18,14)=6402373705728000/(87178291200·24)=3060

C(18,14)=3060; choose y 14 times

3060·x^4·y^14

choose exactly 15 of 18 factors to contribute y

C(18,15)=6402373705728000/(1307674368000·6)=816

C(18,15)=816; choose y 15 times

816·x^3·y^15

choose exactly 16 of 18 factors to contribute y

C(18,16)=6402373705728000/(20922789888000·2)=153

C(18,16)=153; choose y 16 times

153·x^2·y^16

choose exactly 17 of 18 factors to contribute y

C(18,17)=6402373705728000/(355687428096000·1)=18

C(18,17)=18; choose y 17 times

18·x^1·y^17

choose exactly 18 of 18 factors to contribute y

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

C(18,18)=1; choose y 18 times

1·x^0·y^18

all k=0 through 18 have been accounted for

(x+y)^18=1·x^18·y^0 + 18·x^17·y^1 + 153·x^16·y^2 + 816·x^15·y^3 + 3060·x^14·y^4 + 8568·x^13·y^5 + 18564·x^12·y^6 + 31824·x^11·y^7 + 43758·x^10·y^8 + 48620·x^9·y^9 + 43758·x^8·y^10 + 31824·x^7·y^11 + 18564·x^6·y^12 + 8568·x^5·y^13 + 3060·x^4·y^14 + 816·x^3·y^15 + 153·x^2·y^16 + 18·x^1·y^17 + 1·x^0·y^18

Reverse checks

• S040 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• 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

(18, [[0, 1, 18, 0], [1, 18, 17, 1], [2, 153, 16, 2], [3, 816, 15, 3], [4, 3060, 14, 4], [5, 8568, 13, 5], [6, 18564, 12, 6], [7, 31824, 11, 7], [8, 43758, 10, 8], [9, 48620, 9, 9], [10, 43758, 8, 10], [11, 31824, 7, 11], [12, 18564, 6, 12], [13, 8568, 5, 13], [14, 3060, 4, 14], [15, 816, 3, 15], [16, 153, 2, 16], [17, 18, 1, 17], [18, 1, 0, 18]], 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.