Question 93: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=17, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=17, 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=17≥0; x,y commute

(x+y)^17 is a finite product of 17 factors

choose exactly 0 of 17 factors to contribute y

C(17,0)=355687428096000/(1·355687428096000)=1

C(17,0)=1; choose y 0 times

1·x^17·y^0

choose exactly 1 of 17 factors to contribute y

C(17,1)=355687428096000/(1·20922789888000)=17

C(17,1)=17; choose y 1 times

17·x^16·y^1

choose exactly 2 of 17 factors to contribute y

C(17,2)=355687428096000/(2·1307674368000)=136

C(17,2)=136; choose y 2 times

136·x^15·y^2

choose exactly 3 of 17 factors to contribute y

C(17,3)=355687428096000/(6·87178291200)=680

C(17,3)=680; choose y 3 times

680·x^14·y^3

choose exactly 4 of 17 factors to contribute y

C(17,4)=355687428096000/(24·6227020800)=2380

C(17,4)=2380; choose y 4 times

2380·x^13·y^4

choose exactly 5 of 17 factors to contribute y

C(17,5)=355687428096000/(120·479001600)=6188

C(17,5)=6188; choose y 5 times

6188·x^12·y^5

choose exactly 6 of 17 factors to contribute y

C(17,6)=355687428096000/(720·39916800)=12376

C(17,6)=12376; choose y 6 times

12376·x^11·y^6

choose exactly 7 of 17 factors to contribute y

C(17,7)=355687428096000/(5040·3628800)=19448

C(17,7)=19448; choose y 7 times

19448·x^10·y^7

choose exactly 8 of 17 factors to contribute y

C(17,8)=355687428096000/(40320·362880)=24310

C(17,8)=24310; choose y 8 times

24310·x^9·y^8

choose exactly 9 of 17 factors to contribute y

C(17,9)=355687428096000/(362880·40320)=24310

C(17,9)=24310; choose y 9 times

24310·x^8·y^9

choose exactly 10 of 17 factors to contribute y

C(17,10)=355687428096000/(3628800·5040)=19448

C(17,10)=19448; choose y 10 times

19448·x^7·y^10

choose exactly 11 of 17 factors to contribute y

C(17,11)=355687428096000/(39916800·720)=12376

C(17,11)=12376; choose y 11 times

12376·x^6·y^11

choose exactly 12 of 17 factors to contribute y

C(17,12)=355687428096000/(479001600·120)=6188

C(17,12)=6188; choose y 12 times

6188·x^5·y^12

choose exactly 13 of 17 factors to contribute y

C(17,13)=355687428096000/(6227020800·24)=2380

C(17,13)=2380; choose y 13 times

2380·x^4·y^13

choose exactly 14 of 17 factors to contribute y

C(17,14)=355687428096000/(87178291200·6)=680

C(17,14)=680; choose y 14 times

680·x^3·y^14

choose exactly 15 of 17 factors to contribute y

C(17,15)=355687428096000/(1307674368000·2)=136

C(17,15)=136; choose y 15 times

136·x^2·y^15

choose exactly 16 of 17 factors to contribute y

C(17,16)=355687428096000/(20922789888000·1)=17

C(17,16)=17; choose y 16 times

17·x^1·y^16

choose exactly 17 of 17 factors to contribute y

C(17,17)=355687428096000/(355687428096000·1)=1

C(17,17)=1; choose y 17 times

1·x^0·y^17

all k=0 through 17 have been accounted for

(x+y)^17=1·x^17·y^0 + 17·x^16·y^1 + 136·x^15·y^2 + 680·x^14·y^3 + 2380·x^13·y^4 + 6188·x^12·y^5 + 12376·x^11·y^6 + 19448·x^10·y^7 + 24310·x^9·y^8 + 24310·x^8·y^9 + 19448·x^7·y^10 + 12376·x^6·y^11 + 6188·x^5·y^12 + 2380·x^4·y^13 + 680·x^3·y^14 + 136·x^2·y^15 + 17·x^1·y^16 + 1·x^0·y^17

Reverse checks

• S038 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• S037 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S036 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• 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

(17, [[0, 1, 17, 0], [1, 17, 16, 1], [2, 136, 15, 2], [3, 680, 14, 3], [4, 2380, 13, 4], [5, 6188, 12, 5], [6, 12376, 11, 6], [7, 19448, 10, 7], [8, 24310, 9, 8], [9, 24310, 8, 9], [10, 19448, 7, 10], [11, 12376, 6, 11], [12, 6188, 5, 12], [13, 2380, 4, 13], [14, 680, 3, 14], [15, 136, 2, 15], [16, 17, 1, 16], [17, 1, 0, 17]], 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.