Question 90: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=14, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=14, 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=14≥0; x,y commute

(x+y)^14 is a finite product of 14 factors

choose exactly 0 of 14 factors to contribute y

C(14,0)=87178291200/(1·87178291200)=1

C(14,0)=1; choose y 0 times

1·x^14·y^0

choose exactly 1 of 14 factors to contribute y

C(14,1)=87178291200/(1·6227020800)=14

C(14,1)=14; choose y 1 times

14·x^13·y^1

choose exactly 2 of 14 factors to contribute y

C(14,2)=87178291200/(2·479001600)=91

C(14,2)=91; choose y 2 times

91·x^12·y^2

choose exactly 3 of 14 factors to contribute y

C(14,3)=87178291200/(6·39916800)=364

C(14,3)=364; choose y 3 times

364·x^11·y^3

choose exactly 4 of 14 factors to contribute y

C(14,4)=87178291200/(24·3628800)=1001

C(14,4)=1001; choose y 4 times

1001·x^10·y^4

choose exactly 5 of 14 factors to contribute y

C(14,5)=87178291200/(120·362880)=2002

C(14,5)=2002; choose y 5 times

2002·x^9·y^5

choose exactly 6 of 14 factors to contribute y

C(14,6)=87178291200/(720·40320)=3003

C(14,6)=3003; choose y 6 times

3003·x^8·y^6

choose exactly 7 of 14 factors to contribute y

C(14,7)=87178291200/(5040·5040)=3432

C(14,7)=3432; choose y 7 times

3432·x^7·y^7

choose exactly 8 of 14 factors to contribute y

C(14,8)=87178291200/(40320·720)=3003

C(14,8)=3003; choose y 8 times

3003·x^6·y^8

choose exactly 9 of 14 factors to contribute y

C(14,9)=87178291200/(362880·120)=2002

C(14,9)=2002; choose y 9 times

2002·x^5·y^9

choose exactly 10 of 14 factors to contribute y

C(14,10)=87178291200/(3628800·24)=1001

C(14,10)=1001; choose y 10 times

1001·x^4·y^10

choose exactly 11 of 14 factors to contribute y

C(14,11)=87178291200/(39916800·6)=364

C(14,11)=364; choose y 11 times

364·x^3·y^11

choose exactly 12 of 14 factors to contribute y

C(14,12)=87178291200/(479001600·2)=91

C(14,12)=91; choose y 12 times

91·x^2·y^12

choose exactly 13 of 14 factors to contribute y

C(14,13)=87178291200/(6227020800·1)=14

C(14,13)=14; choose y 13 times

14·x^1·y^13

choose exactly 14 of 14 factors to contribute y

C(14,14)=87178291200/(87178291200·1)=1

C(14,14)=1; choose y 14 times

1·x^0·y^14

all k=0 through 14 have been accounted for

(x+y)^14=1·x^14·y^0 + 14·x^13·y^1 + 91·x^12·y^2 + 364·x^11·y^3 + 1001·x^10·y^4 + 2002·x^9·y^5 + 3003·x^8·y^6 + 3432·x^7·y^7 + 3003·x^6·y^8 + 2002·x^5·y^9 + 1001·x^4·y^10 + 364·x^3·y^11 + 91·x^2·y^12 + 14·x^1·y^13 + 1·x^0·y^14

Reverse checks

• S032 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• S031 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S030 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S029 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S028 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• 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

(14, [[0, 1, 14, 0], [1, 14, 13, 1], [2, 91, 12, 2], [3, 364, 11, 3], [4, 1001, 10, 4], [5, 2002, 9, 5], [6, 3003, 8, 6], [7, 3432, 7, 7], [8, 3003, 6, 8], [9, 2002, 5, 9], [10, 1001, 4, 10], [11, 364, 3, 11], [12, 91, 2, 12], [13, 14, 1, 13], [14, 1, 0, 14]], 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.