Question 81: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=5, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=5, 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=5≥0; x,y commute

(x+y)^5 is a finite product of 5 factors

choose exactly 0 of 5 factors to contribute y

C(5,0)=120/(1·120)=1

C(5,0)=1; choose y 0 times

1·x^5·y^0

choose exactly 1 of 5 factors to contribute y

C(5,1)=120/(1·24)=5

C(5,1)=5; choose y 1 times

5·x^4·y^1

choose exactly 2 of 5 factors to contribute y

C(5,2)=120/(2·6)=10

C(5,2)=10; choose y 2 times

10·x^3·y^2

choose exactly 3 of 5 factors to contribute y

C(5,3)=120/(6·2)=10

C(5,3)=10; choose y 3 times

10·x^2·y^3

choose exactly 4 of 5 factors to contribute y

C(5,4)=120/(24·1)=5

C(5,4)=5; choose y 4 times

5·x^1·y^4

choose exactly 5 of 5 factors to contribute y

C(5,5)=120/(120·1)=1

C(5,5)=1; choose y 5 times

1·x^0·y^5

all k=0 through 5 have been accounted for

(x+y)^5=1·x^5·y^0 + 5·x^4·y^1 + 10·x^3·y^2 + 10·x^2·y^3 + 5·x^1·y^4 + 1·x^0·y^5

Reverse checks

• S014 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• 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

(5, [[0, 1, 5, 0], [1, 5, 4, 1], [2, 10, 3, 2], [3, 10, 2, 3], [4, 5, 1, 4], [5, 1, 0, 5]], 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.