Question 100: The Finite Expansion Includes Every k from Zero through n

Original question and givens

Use natural exponent n=24, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).

• G001 — Use natural exponent n=24, 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=24≥0; x,y commute

(x+y)^24 is a finite product of 24 factors

choose exactly 0 of 24 factors to contribute y

C(24,0)=620448401733239439360000/(1·620448401733239439360000)=1

C(24,0)=1; choose y 0 times

1·x^24·y^0

choose exactly 1 of 24 factors to contribute y

C(24,1)=620448401733239439360000/(1·25852016738884976640000)=24

C(24,1)=24; choose y 1 times

24·x^23·y^1

choose exactly 2 of 24 factors to contribute y

C(24,2)=620448401733239439360000/(2·1124000727777607680000)=276

C(24,2)=276; choose y 2 times

276·x^22·y^2

choose exactly 3 of 24 factors to contribute y

C(24,3)=620448401733239439360000/(6·51090942171709440000)=2024

C(24,3)=2024; choose y 3 times

2024·x^21·y^3

choose exactly 4 of 24 factors to contribute y

C(24,4)=620448401733239439360000/(24·2432902008176640000)=10626

C(24,4)=10626; choose y 4 times

10626·x^20·y^4

choose exactly 5 of 24 factors to contribute y

C(24,5)=620448401733239439360000/(120·121645100408832000)=42504

C(24,5)=42504; choose y 5 times

42504·x^19·y^5

choose exactly 6 of 24 factors to contribute y

C(24,6)=620448401733239439360000/(720·6402373705728000)=134596

C(24,6)=134596; choose y 6 times

134596·x^18·y^6

choose exactly 7 of 24 factors to contribute y

C(24,7)=620448401733239439360000/(5040·355687428096000)=346104

C(24,7)=346104; choose y 7 times

346104·x^17·y^7

choose exactly 8 of 24 factors to contribute y

C(24,8)=620448401733239439360000/(40320·20922789888000)=735471

C(24,8)=735471; choose y 8 times

735471·x^16·y^8

choose exactly 9 of 24 factors to contribute y

C(24,9)=620448401733239439360000/(362880·1307674368000)=1307504

C(24,9)=1307504; choose y 9 times

1307504·x^15·y^9

choose exactly 10 of 24 factors to contribute y

C(24,10)=620448401733239439360000/(3628800·87178291200)=1961256

C(24,10)=1961256; choose y 10 times

1961256·x^14·y^10

choose exactly 11 of 24 factors to contribute y

C(24,11)=620448401733239439360000/(39916800·6227020800)=2496144

C(24,11)=2496144; choose y 11 times

2496144·x^13·y^11

choose exactly 12 of 24 factors to contribute y

C(24,12)=620448401733239439360000/(479001600·479001600)=2704156

C(24,12)=2704156; choose y 12 times

2704156·x^12·y^12

choose exactly 13 of 24 factors to contribute y

C(24,13)=620448401733239439360000/(6227020800·39916800)=2496144

C(24,13)=2496144; choose y 13 times

2496144·x^11·y^13

choose exactly 14 of 24 factors to contribute y

C(24,14)=620448401733239439360000/(87178291200·3628800)=1961256

C(24,14)=1961256; choose y 14 times

1961256·x^10·y^14

choose exactly 15 of 24 factors to contribute y

C(24,15)=620448401733239439360000/(1307674368000·362880)=1307504

C(24,15)=1307504; choose y 15 times

1307504·x^9·y^15

choose exactly 16 of 24 factors to contribute y

C(24,16)=620448401733239439360000/(20922789888000·40320)=735471

C(24,16)=735471; choose y 16 times

735471·x^8·y^16

choose exactly 17 of 24 factors to contribute y

C(24,17)=620448401733239439360000/(355687428096000·5040)=346104

C(24,17)=346104; choose y 17 times

346104·x^7·y^17

choose exactly 18 of 24 factors to contribute y

C(24,18)=620448401733239439360000/(6402373705728000·720)=134596

C(24,18)=134596; choose y 18 times

134596·x^6·y^18

choose exactly 19 of 24 factors to contribute y

C(24,19)=620448401733239439360000/(121645100408832000·120)=42504

C(24,19)=42504; choose y 19 times

42504·x^5·y^19

choose exactly 20 of 24 factors to contribute y

C(24,20)=620448401733239439360000/(2432902008176640000·24)=10626

C(24,20)=10626; choose y 20 times

10626·x^4·y^20

choose exactly 21 of 24 factors to contribute y

C(24,21)=620448401733239439360000/(51090942171709440000·6)=2024

C(24,21)=2024; choose y 21 times

2024·x^3·y^21

choose exactly 22 of 24 factors to contribute y

C(24,22)=620448401733239439360000/(1124000727777607680000·2)=276

C(24,22)=276; choose y 22 times

276·x^2·y^22

choose exactly 23 of 24 factors to contribute y

C(24,23)=620448401733239439360000/(25852016738884976640000·1)=24

C(24,23)=24; choose y 23 times

24·x^1·y^23

choose exactly 24 of 24 factors to contribute y

C(24,24)=620448401733239439360000/(620448401733239439360000·1)=1

C(24,24)=1; choose y 24 times

1·x^0·y^24

all k=0 through 24 have been accounted for

(x+y)^24=1·x^24·y^0 + 24·x^23·y^1 + 276·x^22·y^2 + 2024·x^21·y^3 + 10626·x^20·y^4 + 42504·x^19·y^5 + 134596·x^18·y^6 + 346104·x^17·y^7 + 735471·x^16·y^8 + 1307504·x^15·y^9 + 1961256·x^14·y^10 + 2496144·x^13·y^11 + 2704156·x^12·y^12 + 2496144·x^11·y^13 + 1961256·x^10·y^14 + 1307504·x^9·y^15 + 735471·x^8·y^16 + 346104·x^7·y^17 + 134596·x^6·y^18 + 42504·x^5·y^19 + 10626·x^4·y^20 + 2024·x^3·y^21 + 276·x^2·y^22 + 24·x^1·y^23 + 1·x^0·y^24

Reverse checks

• S052 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.
• S051 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.
• S050 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.
• 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

(24, [[0, 1, 24, 0], [1, 24, 23, 1], [2, 276, 22, 2], [3, 2024, 21, 3], [4, 10626, 20, 4], [5, 42504, 19, 5], [6, 134596, 18, 6], [7, 346104, 17, 7], [8, 735471, 16, 8], [9, 1307504, 15, 9], [10, 1961256, 14, 10], [11, 2496144, 13, 11], [12, 2704156, 12, 12], [13, 2496144, 11, 13], [14, 1961256, 10, 14], [15, 1307504, 9, 15], [16, 735471, 8, 16], [17, 346104, 7, 17], [18, 134596, 6, 18], [19, 42504, 5, 19], [20, 10626, 4, 20], [21, 2024, 3, 21], [22, 276, 2, 22], [23, 24, 1, 23], [24, 1, 0, 24]], 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.