Question 80: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=4, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=4, 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=4≥0; x,y commute

(x+y)^4 is a finite product of 4 factors

choose exactly 0 of 4 factors to contribute y

C(4,0)=24/(1·24)=1

C(4,0)=1; choose y 0 times

1·x^4·y^0

choose exactly 1 of 4 factors to contribute y

C(4,1)=24/(1·6)=4

C(4,1)=4; choose y 1 times

4·x^3·y^1

choose exactly 2 of 4 factors to contribute y

C(4,2)=24/(2·2)=6

C(4,2)=6; choose y 2 times

6·x^2·y^2

choose exactly 3 of 4 factors to contribute y

C(4,3)=24/(6·1)=4

C(4,3)=4; choose y 3 times

4·x^1·y^3

choose exactly 4 of 4 factors to contribute y

C(4,4)=24/(24·1)=1

C(4,4)=1; choose y 4 times

1·x^0·y^4

all k=0 through 4 have been accounted for

(x+y)^4=1·x^4·y^0 + 4·x^3·y^1 + 6·x^2·y^2 + 4·x^1·y^3 + 1·x^0·y^4

Reverse checks

• S012 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• 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

(4, [[0, 1, 4, 0], [1, 4, 3, 1], [2, 6, 2, 2], [3, 4, 1, 3], [4, 1, 0, 4]], 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.