Question 92: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=16, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=16, 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=16≥0; x,y commute

(x+y)^16 is a finite product of 16 factors

choose exactly 0 of 16 factors to contribute y

C(16,0)=20922789888000/(1·20922789888000)=1

C(16,0)=1; choose y 0 times

1·x^16·y^0

choose exactly 1 of 16 factors to contribute y

C(16,1)=20922789888000/(1·1307674368000)=16

C(16,1)=16; choose y 1 times

16·x^15·y^1

choose exactly 2 of 16 factors to contribute y

C(16,2)=20922789888000/(2·87178291200)=120

C(16,2)=120; choose y 2 times

120·x^14·y^2

choose exactly 3 of 16 factors to contribute y

C(16,3)=20922789888000/(6·6227020800)=560

C(16,3)=560; choose y 3 times

560·x^13·y^3

choose exactly 4 of 16 factors to contribute y

C(16,4)=20922789888000/(24·479001600)=1820

C(16,4)=1820; choose y 4 times

1820·x^12·y^4

choose exactly 5 of 16 factors to contribute y

C(16,5)=20922789888000/(120·39916800)=4368

C(16,5)=4368; choose y 5 times

4368·x^11·y^5

choose exactly 6 of 16 factors to contribute y

C(16,6)=20922789888000/(720·3628800)=8008

C(16,6)=8008; choose y 6 times

8008·x^10·y^6

choose exactly 7 of 16 factors to contribute y

C(16,7)=20922789888000/(5040·362880)=11440

C(16,7)=11440; choose y 7 times

11440·x^9·y^7

choose exactly 8 of 16 factors to contribute y

C(16,8)=20922789888000/(40320·40320)=12870

C(16,8)=12870; choose y 8 times

12870·x^8·y^8

choose exactly 9 of 16 factors to contribute y

C(16,9)=20922789888000/(362880·5040)=11440

C(16,9)=11440; choose y 9 times

11440·x^7·y^9

choose exactly 10 of 16 factors to contribute y

C(16,10)=20922789888000/(3628800·720)=8008

C(16,10)=8008; choose y 10 times

8008·x^6·y^10

choose exactly 11 of 16 factors to contribute y

C(16,11)=20922789888000/(39916800·120)=4368

C(16,11)=4368; choose y 11 times

4368·x^5·y^11

choose exactly 12 of 16 factors to contribute y

C(16,12)=20922789888000/(479001600·24)=1820

C(16,12)=1820; choose y 12 times

1820·x^4·y^12

choose exactly 13 of 16 factors to contribute y

C(16,13)=20922789888000/(6227020800·6)=560

C(16,13)=560; choose y 13 times

560·x^3·y^13

choose exactly 14 of 16 factors to contribute y

C(16,14)=20922789888000/(87178291200·2)=120

C(16,14)=120; choose y 14 times

120·x^2·y^14

choose exactly 15 of 16 factors to contribute y

C(16,15)=20922789888000/(1307674368000·1)=16

C(16,15)=16; choose y 15 times

16·x^1·y^15

choose exactly 16 of 16 factors to contribute y

C(16,16)=20922789888000/(20922789888000·1)=1

C(16,16)=1; choose y 16 times

1·x^0·y^16

all k=0 through 16 have been accounted for

(x+y)^16=1·x^16·y^0 + 16·x^15·y^1 + 120·x^14·y^2 + 560·x^13·y^3 + 1820·x^12·y^4 + 4368·x^11·y^5 + 8008·x^10·y^6 + 11440·x^9·y^7 + 12870·x^8·y^8 + 11440·x^7·y^9 + 8008·x^6·y^10 + 4368·x^5·y^11 + 1820·x^4·y^12 + 560·x^3·y^13 + 120·x^2·y^14 + 16·x^1·y^15 + 1·x^0·y^16

Reverse checks

• S036 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• S035 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S034 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S033 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S032 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• 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

(16, [[0, 1, 16, 0], [1, 16, 15, 1], [2, 120, 14, 2], [3, 560, 13, 3], [4, 1820, 12, 4], [5, 4368, 11, 5], [6, 8008, 10, 6], [7, 11440, 9, 7], [8, 12870, 8, 8], [9, 11440, 7, 9], [10, 8008, 6, 10], [11, 4368, 5, 11], [12, 1820, 4, 12], [13, 560, 3, 13], [14, 120, 2, 14], [15, 16, 1, 15], [16, 1, 0, 16]], 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.