Question 88: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=12, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=12, 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=12≥0; x,y commute

(x+y)^12 is a finite product of 12 factors

choose exactly 0 of 12 factors to contribute y

C(12,0)=479001600/(1·479001600)=1

C(12,0)=1; choose y 0 times

1·x^12·y^0

choose exactly 1 of 12 factors to contribute y

C(12,1)=479001600/(1·39916800)=12

C(12,1)=12; choose y 1 times

12·x^11·y^1

choose exactly 2 of 12 factors to contribute y

C(12,2)=479001600/(2·3628800)=66

C(12,2)=66; choose y 2 times

66·x^10·y^2

choose exactly 3 of 12 factors to contribute y

C(12,3)=479001600/(6·362880)=220

C(12,3)=220; choose y 3 times

220·x^9·y^3

choose exactly 4 of 12 factors to contribute y

C(12,4)=479001600/(24·40320)=495

C(12,4)=495; choose y 4 times

495·x^8·y^4

choose exactly 5 of 12 factors to contribute y

C(12,5)=479001600/(120·5040)=792

C(12,5)=792; choose y 5 times

792·x^7·y^5

choose exactly 6 of 12 factors to contribute y

C(12,6)=479001600/(720·720)=924

C(12,6)=924; choose y 6 times

924·x^6·y^6

choose exactly 7 of 12 factors to contribute y

C(12,7)=479001600/(5040·120)=792

C(12,7)=792; choose y 7 times

792·x^5·y^7

choose exactly 8 of 12 factors to contribute y

C(12,8)=479001600/(40320·24)=495

C(12,8)=495; choose y 8 times

495·x^4·y^8

choose exactly 9 of 12 factors to contribute y

C(12,9)=479001600/(362880·6)=220

C(12,9)=220; choose y 9 times

220·x^3·y^9

choose exactly 10 of 12 factors to contribute y

C(12,10)=479001600/(3628800·2)=66

C(12,10)=66; choose y 10 times

66·x^2·y^10

choose exactly 11 of 12 factors to contribute y

C(12,11)=479001600/(39916800·1)=12

C(12,11)=12; choose y 11 times

12·x^1·y^11

choose exactly 12 of 12 factors to contribute y

C(12,12)=479001600/(479001600·1)=1

C(12,12)=1; choose y 12 times

1·x^0·y^12

all k=0 through 12 have been accounted for

(x+y)^12=1·x^12·y^0 + 12·x^11·y^1 + 66·x^10·y^2 + 220·x^9·y^3 + 495·x^8·y^4 + 792·x^7·y^5 + 924·x^6·y^6 + 792·x^5·y^7 + 495·x^4·y^8 + 220·x^3·y^9 + 66·x^2·y^10 + 12·x^1·y^11 + 1·x^0·y^12

Reverse checks

• S028 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• S027 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S026 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• 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

(12, [[0, 1, 12, 0], [1, 12, 11, 1], [2, 66, 10, 2], [3, 220, 9, 3], [4, 495, 8, 4], [5, 792, 7, 5], [6, 924, 6, 6], [7, 792, 5, 7], [8, 495, 4, 8], [9, 220, 3, 9], [10, 66, 2, 10], [11, 12, 1, 11], [12, 1, 0, 12]], 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.