Question 85: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=9, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=9, 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=9≥0; x,y commute

(x+y)^9 is a finite product of 9 factors

choose exactly 0 of 9 factors to contribute y

C(9,0)=362880/(1·362880)=1

C(9,0)=1; choose y 0 times

1·x^9·y^0

choose exactly 1 of 9 factors to contribute y

C(9,1)=362880/(1·40320)=9

C(9,1)=9; choose y 1 times

9·x^8·y^1

choose exactly 2 of 9 factors to contribute y

C(9,2)=362880/(2·5040)=36

C(9,2)=36; choose y 2 times

36·x^7·y^2

choose exactly 3 of 9 factors to contribute y

C(9,3)=362880/(6·720)=84

C(9,3)=84; choose y 3 times

84·x^6·y^3

choose exactly 4 of 9 factors to contribute y

C(9,4)=362880/(24·120)=126

C(9,4)=126; choose y 4 times

126·x^5·y^4

choose exactly 5 of 9 factors to contribute y

C(9,5)=362880/(120·24)=126

C(9,5)=126; choose y 5 times

126·x^4·y^5

choose exactly 6 of 9 factors to contribute y

C(9,6)=362880/(720·6)=84

C(9,6)=84; choose y 6 times

84·x^3·y^6

choose exactly 7 of 9 factors to contribute y

C(9,7)=362880/(5040·2)=36

C(9,7)=36; choose y 7 times

36·x^2·y^7

choose exactly 8 of 9 factors to contribute y

C(9,8)=362880/(40320·1)=9

C(9,8)=9; choose y 8 times

9·x^1·y^8

choose exactly 9 of 9 factors to contribute y

C(9,9)=362880/(362880·1)=1

C(9,9)=1; choose y 9 times

1·x^0·y^9

all k=0 through 9 have been accounted for

(x+y)^9=1·x^9·y^0 + 9·x^8·y^1 + 36·x^7·y^2 + 84·x^6·y^3 + 126·x^5·y^4 + 126·x^4·y^5 + 84·x^3·y^6 + 36·x^2·y^7 + 9·x^1·y^8 + 1·x^0·y^9

Reverse checks

• S022 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• 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

(9, [[0, 1, 9, 0], [1, 9, 8, 1], [2, 36, 7, 2], [3, 84, 6, 3], [4, 126, 5, 4], [5, 126, 4, 5], [6, 84, 3, 6], [7, 36, 2, 7], [8, 9, 1, 8], [9, 1, 0, 9]], 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.