Question 91: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=15, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=15, 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=15≥0; x,y commute

(x+y)^15 is a finite product of 15 factors

choose exactly 0 of 15 factors to contribute y

C(15,0)=1307674368000/(1·1307674368000)=1

C(15,0)=1; choose y 0 times

1·x^15·y^0

choose exactly 1 of 15 factors to contribute y

C(15,1)=1307674368000/(1·87178291200)=15

C(15,1)=15; choose y 1 times

15·x^14·y^1

choose exactly 2 of 15 factors to contribute y

C(15,2)=1307674368000/(2·6227020800)=105

C(15,2)=105; choose y 2 times

105·x^13·y^2

choose exactly 3 of 15 factors to contribute y

C(15,3)=1307674368000/(6·479001600)=455

C(15,3)=455; choose y 3 times

455·x^12·y^3

choose exactly 4 of 15 factors to contribute y

C(15,4)=1307674368000/(24·39916800)=1365

C(15,4)=1365; choose y 4 times

1365·x^11·y^4

choose exactly 5 of 15 factors to contribute y

C(15,5)=1307674368000/(120·3628800)=3003

C(15,5)=3003; choose y 5 times

3003·x^10·y^5

choose exactly 6 of 15 factors to contribute y

C(15,6)=1307674368000/(720·362880)=5005

C(15,6)=5005; choose y 6 times

5005·x^9·y^6

choose exactly 7 of 15 factors to contribute y

C(15,7)=1307674368000/(5040·40320)=6435

C(15,7)=6435; choose y 7 times

6435·x^8·y^7

choose exactly 8 of 15 factors to contribute y

C(15,8)=1307674368000/(40320·5040)=6435

C(15,8)=6435; choose y 8 times

6435·x^7·y^8

choose exactly 9 of 15 factors to contribute y

C(15,9)=1307674368000/(362880·720)=5005

C(15,9)=5005; choose y 9 times

5005·x^6·y^9

choose exactly 10 of 15 factors to contribute y

C(15,10)=1307674368000/(3628800·120)=3003

C(15,10)=3003; choose y 10 times

3003·x^5·y^10

choose exactly 11 of 15 factors to contribute y

C(15,11)=1307674368000/(39916800·24)=1365

C(15,11)=1365; choose y 11 times

1365·x^4·y^11

choose exactly 12 of 15 factors to contribute y

C(15,12)=1307674368000/(479001600·6)=455

C(15,12)=455; choose y 12 times

455·x^3·y^12

choose exactly 13 of 15 factors to contribute y

C(15,13)=1307674368000/(6227020800·2)=105

C(15,13)=105; choose y 13 times

105·x^2·y^13

choose exactly 14 of 15 factors to contribute y

C(15,14)=1307674368000/(87178291200·1)=15

C(15,14)=15; choose y 14 times

15·x^1·y^14

choose exactly 15 of 15 factors to contribute y

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

C(15,15)=1; choose y 15 times

1·x^0·y^15

all k=0 through 15 have been accounted for

(x+y)^15=1·x^15·y^0 + 15·x^14·y^1 + 105·x^13·y^2 + 455·x^12·y^3 + 1365·x^11·y^4 + 3003·x^10·y^5 + 5005·x^9·y^6 + 6435·x^8·y^7 + 6435·x^7·y^8 + 5005·x^6·y^9 + 3003·x^5·y^10 + 1365·x^4·y^11 + 455·x^3·y^12 + 105·x^2·y^13 + 15·x^1·y^14 + 1·x^0·y^15

Reverse checks

• S034 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• 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

(15, [[0, 1, 15, 0], [1, 15, 14, 1], [2, 105, 13, 2], [3, 455, 12, 3], [4, 1365, 11, 4], [5, 3003, 10, 5], [6, 5005, 9, 6], [7, 6435, 8, 7], [8, 6435, 7, 8], [9, 5005, 6, 9], [10, 3003, 5, 10], [11, 1365, 4, 11], [12, 455, 3, 12], [13, 105, 2, 13], [14, 15, 1, 14], [15, 1, 0, 15]], 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.