Question 89: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=13, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=13, 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=13≥0; x,y commute

(x+y)^13 is a finite product of 13 factors

choose exactly 0 of 13 factors to contribute y

C(13,0)=6227020800/(1·6227020800)=1

C(13,0)=1; choose y 0 times

1·x^13·y^0

choose exactly 1 of 13 factors to contribute y

C(13,1)=6227020800/(1·479001600)=13

C(13,1)=13; choose y 1 times

13·x^12·y^1

choose exactly 2 of 13 factors to contribute y

C(13,2)=6227020800/(2·39916800)=78

C(13,2)=78; choose y 2 times

78·x^11·y^2

choose exactly 3 of 13 factors to contribute y

C(13,3)=6227020800/(6·3628800)=286

C(13,3)=286; choose y 3 times

286·x^10·y^3

choose exactly 4 of 13 factors to contribute y

C(13,4)=6227020800/(24·362880)=715

C(13,4)=715; choose y 4 times

715·x^9·y^4

choose exactly 5 of 13 factors to contribute y

C(13,5)=6227020800/(120·40320)=1287

C(13,5)=1287; choose y 5 times

1287·x^8·y^5

choose exactly 6 of 13 factors to contribute y

C(13,6)=6227020800/(720·5040)=1716

C(13,6)=1716; choose y 6 times

1716·x^7·y^6

choose exactly 7 of 13 factors to contribute y

C(13,7)=6227020800/(5040·720)=1716

C(13,7)=1716; choose y 7 times

1716·x^6·y^7

choose exactly 8 of 13 factors to contribute y

C(13,8)=6227020800/(40320·120)=1287

C(13,8)=1287; choose y 8 times

1287·x^5·y^8

choose exactly 9 of 13 factors to contribute y

C(13,9)=6227020800/(362880·24)=715

C(13,9)=715; choose y 9 times

715·x^4·y^9

choose exactly 10 of 13 factors to contribute y

C(13,10)=6227020800/(3628800·6)=286

C(13,10)=286; choose y 10 times

286·x^3·y^10

choose exactly 11 of 13 factors to contribute y

C(13,11)=6227020800/(39916800·2)=78

C(13,11)=78; choose y 11 times

78·x^2·y^11

choose exactly 12 of 13 factors to contribute y

C(13,12)=6227020800/(479001600·1)=13

C(13,12)=13; choose y 12 times

13·x^1·y^12

choose exactly 13 of 13 factors to contribute y

C(13,13)=6227020800/(6227020800·1)=1

C(13,13)=1; choose y 13 times

1·x^0·y^13

all k=0 through 13 have been accounted for

(x+y)^13=1·x^13·y^0 + 13·x^12·y^1 + 78·x^11·y^2 + 286·x^10·y^3 + 715·x^9·y^4 + 1287·x^8·y^5 + 1716·x^7·y^6 + 1716·x^6·y^7 + 1287·x^5·y^8 + 715·x^4·y^9 + 286·x^3·y^10 + 78·x^2·y^11 + 13·x^1·y^12 + 1·x^0·y^13

Reverse checks

• S030 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• 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

(13, [[0, 1, 13, 0], [1, 13, 12, 1], [2, 78, 11, 2], [3, 286, 10, 3], [4, 715, 9, 4], [5, 1287, 8, 5], [6, 1716, 7, 6], [7, 1716, 6, 7], [8, 1287, 5, 8], [9, 715, 4, 9], [10, 286, 3, 10], [11, 78, 2, 11], [12, 13, 1, 12], [13, 1, 0, 13]], 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.