Question 87: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=11, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=11, 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=11≥0; x,y commute

(x+y)^11 is a finite product of 11 factors

choose exactly 0 of 11 factors to contribute y

C(11,0)=39916800/(1·39916800)=1

C(11,0)=1; choose y 0 times

1·x^11·y^0

choose exactly 1 of 11 factors to contribute y

C(11,1)=39916800/(1·3628800)=11

C(11,1)=11; choose y 1 times

11·x^10·y^1

choose exactly 2 of 11 factors to contribute y

C(11,2)=39916800/(2·362880)=55

C(11,2)=55; choose y 2 times

55·x^9·y^2

choose exactly 3 of 11 factors to contribute y

C(11,3)=39916800/(6·40320)=165

C(11,3)=165; choose y 3 times

165·x^8·y^3

choose exactly 4 of 11 factors to contribute y

C(11,4)=39916800/(24·5040)=330

C(11,4)=330; choose y 4 times

330·x^7·y^4

choose exactly 5 of 11 factors to contribute y

C(11,5)=39916800/(120·720)=462

C(11,5)=462; choose y 5 times

462·x^6·y^5

choose exactly 6 of 11 factors to contribute y

C(11,6)=39916800/(720·120)=462

C(11,6)=462; choose y 6 times

462·x^5·y^6

choose exactly 7 of 11 factors to contribute y

C(11,7)=39916800/(5040·24)=330

C(11,7)=330; choose y 7 times

330·x^4·y^7

choose exactly 8 of 11 factors to contribute y

C(11,8)=39916800/(40320·6)=165

C(11,8)=165; choose y 8 times

165·x^3·y^8

choose exactly 9 of 11 factors to contribute y

C(11,9)=39916800/(362880·2)=55

C(11,9)=55; choose y 9 times

55·x^2·y^9

choose exactly 10 of 11 factors to contribute y

C(11,10)=39916800/(3628800·1)=11

C(11,10)=11; choose y 10 times

11·x^1·y^10

choose exactly 11 of 11 factors to contribute y

C(11,11)=39916800/(39916800·1)=1

C(11,11)=1; choose y 11 times

1·x^0·y^11

all k=0 through 11 have been accounted for

(x+y)^11=1·x^11·y^0 + 11·x^10·y^1 + 55·x^9·y^2 + 165·x^8·y^3 + 330·x^7·y^4 + 462·x^6·y^5 + 462·x^5·y^6 + 330·x^4·y^7 + 165·x^3·y^8 + 55·x^2·y^9 + 11·x^1·y^10 + 1·x^0·y^11

Reverse checks

• S026 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• S025 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S024 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• 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

(11, [[0, 1, 11, 0], [1, 11, 10, 1], [2, 55, 9, 2], [3, 165, 8, 3], [4, 330, 7, 4], [5, 462, 6, 5], [6, 462, 5, 6], [7, 330, 4, 7], [8, 165, 3, 8], [9, 55, 2, 9], [10, 11, 1, 10], [11, 1, 0, 11]], 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.