Question 99: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=23, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=23, 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=23≥0; x,y commute

(x+y)^23 is a finite product of 23 factors

choose exactly 0 of 23 factors to contribute y

C(23,0)=25852016738884976640000/(1·25852016738884976640000)=1

C(23,0)=1; choose y 0 times

1·x^23·y^0

choose exactly 1 of 23 factors to contribute y

C(23,1)=25852016738884976640000/(1·1124000727777607680000)=23

C(23,1)=23; choose y 1 times

23·x^22·y^1

choose exactly 2 of 23 factors to contribute y

C(23,2)=25852016738884976640000/(2·51090942171709440000)=253

C(23,2)=253; choose y 2 times

253·x^21·y^2

choose exactly 3 of 23 factors to contribute y

C(23,3)=25852016738884976640000/(6·2432902008176640000)=1771

C(23,3)=1771; choose y 3 times

1771·x^20·y^3

choose exactly 4 of 23 factors to contribute y

C(23,4)=25852016738884976640000/(24·121645100408832000)=8855

C(23,4)=8855; choose y 4 times

8855·x^19·y^4

choose exactly 5 of 23 factors to contribute y

C(23,5)=25852016738884976640000/(120·6402373705728000)=33649

C(23,5)=33649; choose y 5 times

33649·x^18·y^5

choose exactly 6 of 23 factors to contribute y

C(23,6)=25852016738884976640000/(720·355687428096000)=100947

C(23,6)=100947; choose y 6 times

100947·x^17·y^6

choose exactly 7 of 23 factors to contribute y

C(23,7)=25852016738884976640000/(5040·20922789888000)=245157

C(23,7)=245157; choose y 7 times

245157·x^16·y^7

choose exactly 8 of 23 factors to contribute y

C(23,8)=25852016738884976640000/(40320·1307674368000)=490314

C(23,8)=490314; choose y 8 times

490314·x^15·y^8

choose exactly 9 of 23 factors to contribute y

C(23,9)=25852016738884976640000/(362880·87178291200)=817190

C(23,9)=817190; choose y 9 times

817190·x^14·y^9

choose exactly 10 of 23 factors to contribute y

C(23,10)=25852016738884976640000/(3628800·6227020800)=1144066

C(23,10)=1144066; choose y 10 times

1144066·x^13·y^10

choose exactly 11 of 23 factors to contribute y

C(23,11)=25852016738884976640000/(39916800·479001600)=1352078

C(23,11)=1352078; choose y 11 times

1352078·x^12·y^11

choose exactly 12 of 23 factors to contribute y

C(23,12)=25852016738884976640000/(479001600·39916800)=1352078

C(23,12)=1352078; choose y 12 times

1352078·x^11·y^12

choose exactly 13 of 23 factors to contribute y

C(23,13)=25852016738884976640000/(6227020800·3628800)=1144066

C(23,13)=1144066; choose y 13 times

1144066·x^10·y^13

choose exactly 14 of 23 factors to contribute y

C(23,14)=25852016738884976640000/(87178291200·362880)=817190

C(23,14)=817190; choose y 14 times

817190·x^9·y^14

choose exactly 15 of 23 factors to contribute y

C(23,15)=25852016738884976640000/(1307674368000·40320)=490314

C(23,15)=490314; choose y 15 times

490314·x^8·y^15

choose exactly 16 of 23 factors to contribute y

C(23,16)=25852016738884976640000/(20922789888000·5040)=245157

C(23,16)=245157; choose y 16 times

245157·x^7·y^16

choose exactly 17 of 23 factors to contribute y

C(23,17)=25852016738884976640000/(355687428096000·720)=100947

C(23,17)=100947; choose y 17 times

100947·x^6·y^17

choose exactly 18 of 23 factors to contribute y

C(23,18)=25852016738884976640000/(6402373705728000·120)=33649

C(23,18)=33649; choose y 18 times

33649·x^5·y^18

choose exactly 19 of 23 factors to contribute y

C(23,19)=25852016738884976640000/(121645100408832000·24)=8855

C(23,19)=8855; choose y 19 times

8855·x^4·y^19

choose exactly 20 of 23 factors to contribute y

C(23,20)=25852016738884976640000/(2432902008176640000·6)=1771

C(23,20)=1771; choose y 20 times

1771·x^3·y^20

choose exactly 21 of 23 factors to contribute y

C(23,21)=25852016738884976640000/(51090942171709440000·2)=253

C(23,21)=253; choose y 21 times

253·x^2·y^21

choose exactly 22 of 23 factors to contribute y

C(23,22)=25852016738884976640000/(1124000727777607680000·1)=23

C(23,22)=23; choose y 22 times

23·x^1·y^22

choose exactly 23 of 23 factors to contribute y

C(23,23)=25852016738884976640000/(25852016738884976640000·1)=1

C(23,23)=1; choose y 23 times

1·x^0·y^23

all k=0 through 23 have been accounted for

(x+y)^23=1·x^23·y^0 + 23·x^22·y^1 + 253·x^21·y^2 + 1771·x^20·y^3 + 8855·x^19·y^4 + 33649·x^18·y^5 + 100947·x^17·y^6 + 245157·x^16·y^7 + 490314·x^15·y^8 + 817190·x^14·y^9 + 1144066·x^13·y^10 + 1352078·x^12·y^11 + 1352078·x^11·y^12 + 1144066·x^10·y^13 + 817190·x^9·y^14 + 490314·x^8·y^15 + 245157·x^7·y^16 + 100947·x^6·y^17 + 33649·x^5·y^18 + 8855·x^4·y^19 + 1771·x^3·y^20 + 253·x^2·y^21 + 23·x^1·y^22 + 1·x^0·y^23

Reverse checks

• S050 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• S049 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S048 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S047 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S046 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S045 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S044 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S043 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S042 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S041 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S040 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• S039 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S038 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• 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

(23, [[0, 1, 23, 0], [1, 23, 22, 1], [2, 253, 21, 2], [3, 1771, 20, 3], [4, 8855, 19, 4], [5, 33649, 18, 5], [6, 100947, 17, 6], [7, 245157, 16, 7], [8, 490314, 15, 8], [9, 817190, 14, 9], [10, 1144066, 13, 10], [11, 1352078, 12, 11], [12, 1352078, 11, 12], [13, 1144066, 10, 13], [14, 817190, 9, 14], [15, 490314, 8, 15], [16, 245157, 7, 16], [17, 100947, 6, 17], [18, 33649, 5, 18], [19, 8855, 4, 19], [20, 1771, 3, 20], [21, 253, 2, 21], [22, 23, 1, 22], [23, 1, 0, 23]], 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.