Question 83: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=7, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=7, 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=7≥0; x,y commute

(x+y)^7 is a finite product of 7 factors

choose exactly 0 of 7 factors to contribute y

C(7,0)=5040/(1·5040)=1

C(7,0)=1; choose y 0 times

1·x^7·y^0

choose exactly 1 of 7 factors to contribute y

C(7,1)=5040/(1·720)=7

C(7,1)=7; choose y 1 times

7·x^6·y^1

choose exactly 2 of 7 factors to contribute y

C(7,2)=5040/(2·120)=21

C(7,2)=21; choose y 2 times

21·x^5·y^2

choose exactly 3 of 7 factors to contribute y

C(7,3)=5040/(6·24)=35

C(7,3)=35; choose y 3 times

35·x^4·y^3

choose exactly 4 of 7 factors to contribute y

C(7,4)=5040/(24·6)=35

C(7,4)=35; choose y 4 times

35·x^3·y^4

choose exactly 5 of 7 factors to contribute y

C(7,5)=5040/(120·2)=21

C(7,5)=21; choose y 5 times

21·x^2·y^5

choose exactly 6 of 7 factors to contribute y

C(7,6)=5040/(720·1)=7

C(7,6)=7; choose y 6 times

7·x^1·y^6

choose exactly 7 of 7 factors to contribute y

C(7,7)=5040/(5040·1)=1

C(7,7)=1; choose y 7 times

1·x^0·y^7

all k=0 through 7 have been accounted for

(x+y)^7=1·x^7·y^0 + 7·x^6·y^1 + 21·x^5·y^2 + 35·x^4·y^3 + 35·x^3·y^4 + 21·x^2·y^5 + 7·x^1·y^6 + 1·x^0·y^7

Reverse checks

• S018 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• 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

(7, [[0, 1, 7, 0], [1, 7, 6, 1], [2, 21, 5, 2], [3, 35, 4, 3], [4, 35, 3, 4], [5, 21, 2, 5], [6, 7, 1, 6], [7, 1, 0, 7]], 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.