Question 84: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=8, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=8, 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=8≥0; x,y commute

(x+y)^8 is a finite product of 8 factors

choose exactly 0 of 8 factors to contribute y

C(8,0)=40320/(1·40320)=1

C(8,0)=1; choose y 0 times

1·x^8·y^0

choose exactly 1 of 8 factors to contribute y

C(8,1)=40320/(1·5040)=8

C(8,1)=8; choose y 1 times

8·x^7·y^1

choose exactly 2 of 8 factors to contribute y

C(8,2)=40320/(2·720)=28

C(8,2)=28; choose y 2 times

28·x^6·y^2

choose exactly 3 of 8 factors to contribute y

C(8,3)=40320/(6·120)=56

C(8,3)=56; choose y 3 times

56·x^5·y^3

choose exactly 4 of 8 factors to contribute y

C(8,4)=40320/(24·24)=70

C(8,4)=70; choose y 4 times

70·x^4·y^4

choose exactly 5 of 8 factors to contribute y

C(8,5)=40320/(120·6)=56

C(8,5)=56; choose y 5 times

56·x^3·y^5

choose exactly 6 of 8 factors to contribute y

C(8,6)=40320/(720·2)=28

C(8,6)=28; choose y 6 times

28·x^2·y^6

choose exactly 7 of 8 factors to contribute y

C(8,7)=40320/(5040·1)=8

C(8,7)=8; choose y 7 times

8·x^1·y^7

choose exactly 8 of 8 factors to contribute y

C(8,8)=40320/(40320·1)=1

C(8,8)=1; choose y 8 times

1·x^0·y^8

all k=0 through 8 have been accounted for

(x+y)^8=1·x^8·y^0 + 8·x^7·y^1 + 28·x^6·y^2 + 56·x^5·y^3 + 70·x^4·y^4 + 56·x^3·y^5 + 28·x^2·y^6 + 8·x^1·y^7 + 1·x^0·y^8

Reverse checks

• S020 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• 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

(8, [[0, 1, 8, 0], [1, 8, 7, 1], [2, 28, 6, 2], [3, 56, 5, 3], [4, 70, 4, 4], [5, 56, 3, 5], [6, 28, 2, 6], [7, 8, 1, 7], [8, 1, 0, 8]], 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.