Question 86: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=10, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=10, 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=10≥0; x,y commute

(x+y)^10 is a finite product of 10 factors

choose exactly 0 of 10 factors to contribute y

C(10,0)=3628800/(1·3628800)=1

C(10,0)=1; choose y 0 times

1·x^10·y^0

choose exactly 1 of 10 factors to contribute y

C(10,1)=3628800/(1·362880)=10

C(10,1)=10; choose y 1 times

10·x^9·y^1

choose exactly 2 of 10 factors to contribute y

C(10,2)=3628800/(2·40320)=45

C(10,2)=45; choose y 2 times

45·x^8·y^2

choose exactly 3 of 10 factors to contribute y

C(10,3)=3628800/(6·5040)=120

C(10,3)=120; choose y 3 times

120·x^7·y^3

choose exactly 4 of 10 factors to contribute y

C(10,4)=3628800/(24·720)=210

C(10,4)=210; choose y 4 times

210·x^6·y^4

choose exactly 5 of 10 factors to contribute y

C(10,5)=3628800/(120·120)=252

C(10,5)=252; choose y 5 times

252·x^5·y^5

choose exactly 6 of 10 factors to contribute y

C(10,6)=3628800/(720·24)=210

C(10,6)=210; choose y 6 times

210·x^4·y^6

choose exactly 7 of 10 factors to contribute y

C(10,7)=3628800/(5040·6)=120

C(10,7)=120; choose y 7 times

120·x^3·y^7

choose exactly 8 of 10 factors to contribute y

C(10,8)=3628800/(40320·2)=45

C(10,8)=45; choose y 8 times

45·x^2·y^8

choose exactly 9 of 10 factors to contribute y

C(10,9)=3628800/(362880·1)=10

C(10,9)=10; choose y 9 times

10·x^1·y^9

choose exactly 10 of 10 factors to contribute y

C(10,10)=3628800/(3628800·1)=1

C(10,10)=1; choose y 10 times

1·x^0·y^10

all k=0 through 10 have been accounted for

(x+y)^10=1·x^10·y^0 + 10·x^9·y^1 + 45·x^8·y^2 + 120·x^7·y^3 + 210·x^6·y^4 + 252·x^5·y^5 + 210·x^4·y^6 + 120·x^3·y^7 + 45·x^2·y^8 + 10·x^1·y^9 + 1·x^0·y^10

Reverse checks

• S024 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• S023 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S022 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• 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

(10, [[0, 1, 10, 0], [1, 10, 9, 1], [2, 45, 8, 2], [3, 120, 7, 3], [4, 210, 6, 4], [5, 252, 5, 5], [6, 210, 4, 6], [7, 120, 3, 7], [8, 45, 2, 8], [9, 10, 1, 9], [10, 1, 0, 10]], 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.