# Question 97: The Finite Expansion Includes Every k from Zero through n [Return to catalogue page 1](https://ocme.evemisslab.com/en/questions/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261109-r05-binomial-theorem-foundations/#q-20261014-advance-20261109-r05-binomial-theorem-0047) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. P250-0097 · q-20261014-advance-20261109-r05-binomial-theorem-0047 ## Original question and givens Use natural exponent n=21, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary). - G001 — Use natural exponent n=21, 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](https://ocme.evemisslab.com/en/mko/mko-binomial-theorem/) ## Step-by-step derivation S001 · impliesBefore n=21≥0; x,y commute After (x+y)^21 is a finite product of 21 factors Conditions: none Choose x or y from each factor; commutativity permits grouping products by the number of y choices. Step identifiers and dependenciesRule: finite_product · Uses: G001 S002 · impliesBefore choose exactly 0 of 21 factors to contribute y After C(21,0)=51090942171709440000/(1·51090942171709440000)=1 Conditions: none Count orderings of 21 positions and divide by reorderings within the 0 y positions and 21 x positions, giving coefficient 1. Step identifiers and dependenciesRule: subset_count · Uses: G001, S001 S003 · impliesBefore C(21,0)=1; choose y 0 times After 1·x^21·y^0 Conditions: none The x exponent is 21-0=21; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S002 S004 · impliesBefore choose exactly 1 of 21 factors to contribute y After C(21,1)=51090942171709440000/(1·2432902008176640000)=21 Conditions: none Count orderings of 21 positions and divide by reorderings within the 1 y positions and 20 x positions, giving coefficient 21. Step identifiers and dependenciesRule: subset_count · Uses: G001, S003 S005 · impliesBefore C(21,1)=21; choose y 1 times After 21·x^20·y^1 Conditions: none The x exponent is 21-1=20; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S004 S006 · impliesBefore choose exactly 2 of 21 factors to contribute y After C(21,2)=51090942171709440000/(2·121645100408832000)=210 Conditions: none Count orderings of 21 positions and divide by reorderings within the 2 y positions and 19 x positions, giving coefficient 210. Step identifiers and dependenciesRule: subset_count · Uses: G001, S005 S007 · impliesBefore C(21,2)=210; choose y 2 times After 210·x^19·y^2 Conditions: none The x exponent is 21-2=19; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S006 S008 · impliesBefore choose exactly 3 of 21 factors to contribute y After C(21,3)=51090942171709440000/(6·6402373705728000)=1330 Conditions: none Count orderings of 21 positions and divide by reorderings within the 3 y positions and 18 x positions, giving coefficient 1330. Step identifiers and dependenciesRule: subset_count · Uses: G001, S007 S009 · impliesBefore C(21,3)=1330; choose y 3 times After 1330·x^18·y^3 Conditions: none The x exponent is 21-3=18; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S008 S010 · impliesBefore choose exactly 4 of 21 factors to contribute y After C(21,4)=51090942171709440000/(24·355687428096000)=5985 Conditions: none Count orderings of 21 positions and divide by reorderings within the 4 y positions and 17 x positions, giving coefficient 5985. Step identifiers and dependenciesRule: subset_count · Uses: G001, S009 S011 · impliesBefore C(21,4)=5985; choose y 4 times After 5985·x^17·y^4 Conditions: none The x exponent is 21-4=17; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S010 S012 · impliesBefore choose exactly 5 of 21 factors to contribute y After C(21,5)=51090942171709440000/(120·20922789888000)=20349 Conditions: none Count orderings of 21 positions and divide by reorderings within the 5 y positions and 16 x positions, giving coefficient 20349. Step identifiers and dependenciesRule: subset_count · Uses: G001, S011 S013 · impliesBefore C(21,5)=20349; choose y 5 times After 20349·x^16·y^5 Conditions: none The x exponent is 21-5=16; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S012 S014 · impliesBefore choose exactly 6 of 21 factors to contribute y After C(21,6)=51090942171709440000/(720·1307674368000)=54264 Conditions: none Count orderings of 21 positions and divide by reorderings within the 6 y positions and 15 x positions, giving coefficient 54264. Step identifiers and dependenciesRule: subset_count · Uses: G001, S013 S015 · impliesBefore C(21,6)=54264; choose y 6 times After 54264·x^15·y^6 Conditions: none The x exponent is 21-6=15; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S014 S016 · impliesBefore choose exactly 7 of 21 factors to contribute y After C(21,7)=51090942171709440000/(5040·87178291200)=116280 Conditions: none Count orderings of 21 positions and divide by reorderings within the 7 y positions and 14 x positions, giving coefficient 116280. Step identifiers and dependenciesRule: subset_count · Uses: G001, S015 S017 · impliesBefore C(21,7)=116280; choose y 7 times After 116280·x^14·y^7 Conditions: none The x exponent is 21-7=14; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S016 S018 · impliesBefore choose exactly 8 of 21 factors to contribute y After C(21,8)=51090942171709440000/(40320·6227020800)=203490 Conditions: none Count orderings of 21 positions and divide by reorderings within the 8 y positions and 13 x positions, giving coefficient 203490. Step identifiers and dependenciesRule: subset_count · Uses: G001, S017 S019 · impliesBefore C(21,8)=203490; choose y 8 times After 203490·x^13·y^8 Conditions: none The x exponent is 21-8=13; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S018 S020 · impliesBefore choose exactly 9 of 21 factors to contribute y After C(21,9)=51090942171709440000/(362880·479001600)=293930 Conditions: none Count orderings of 21 positions and divide by reorderings within the 9 y positions and 12 x positions, giving coefficient 293930. Step identifiers and dependenciesRule: subset_count · Uses: G001, S019 S021 · impliesBefore C(21,9)=293930; choose y 9 times After 293930·x^12·y^9 Conditions: none The x exponent is 21-9=12; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S020 S022 · impliesBefore choose exactly 10 of 21 factors to contribute y After C(21,10)=51090942171709440000/(3628800·39916800)=352716 Conditions: none Count orderings of 21 positions and divide by reorderings within the 10 y positions and 11 x positions, giving coefficient 352716. Step identifiers and dependenciesRule: subset_count · Uses: G001, S021 S023 · impliesBefore C(21,10)=352716; choose y 10 times After 352716·x^11·y^10 Conditions: none The x exponent is 21-10=11; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S022 S024 · impliesBefore choose exactly 11 of 21 factors to contribute y After C(21,11)=51090942171709440000/(39916800·3628800)=352716 Conditions: none Count orderings of 21 positions and divide by reorderings within the 11 y positions and 10 x positions, giving coefficient 352716. Step identifiers and dependenciesRule: subset_count · Uses: G001, S023 S025 · impliesBefore C(21,11)=352716; choose y 11 times After 352716·x^10·y^11 Conditions: none The x exponent is 21-11=10; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S024 S026 · impliesBefore choose exactly 12 of 21 factors to contribute y After C(21,12)=51090942171709440000/(479001600·362880)=293930 Conditions: none Count orderings of 21 positions and divide by reorderings within the 12 y positions and 9 x positions, giving coefficient 293930. Step identifiers and dependenciesRule: subset_count · Uses: G001, S025 S027 · impliesBefore C(21,12)=293930; choose y 12 times After 293930·x^9·y^12 Conditions: none The x exponent is 21-12=9; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S026 S028 · impliesBefore choose exactly 13 of 21 factors to contribute y After C(21,13)=51090942171709440000/(6227020800·40320)=203490 Conditions: none Count orderings of 21 positions and divide by reorderings within the 13 y positions and 8 x positions, giving coefficient 203490. Step identifiers and dependenciesRule: subset_count · Uses: G001, S027 S029 · impliesBefore C(21,13)=203490; choose y 13 times After 203490·x^8·y^13 Conditions: none The x exponent is 21-13=8; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S028 S030 · impliesBefore choose exactly 14 of 21 factors to contribute y After C(21,14)=51090942171709440000/(87178291200·5040)=116280 Conditions: none Count orderings of 21 positions and divide by reorderings within the 14 y positions and 7 x positions, giving coefficient 116280. Step identifiers and dependenciesRule: subset_count · Uses: G001, S029 S031 · impliesBefore C(21,14)=116280; choose y 14 times After 116280·x^7·y^14 Conditions: none The x exponent is 21-14=7; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S030 S032 · impliesBefore choose exactly 15 of 21 factors to contribute y After C(21,15)=51090942171709440000/(1307674368000·720)=54264 Conditions: none Count orderings of 21 positions and divide by reorderings within the 15 y positions and 6 x positions, giving coefficient 54264. Step identifiers and dependenciesRule: subset_count · Uses: G001, S031 S033 · impliesBefore C(21,15)=54264; choose y 15 times After 54264·x^6·y^15 Conditions: none The x exponent is 21-15=6; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S032 S034 · impliesBefore choose exactly 16 of 21 factors to contribute y After C(21,16)=51090942171709440000/(20922789888000·120)=20349 Conditions: none Count orderings of 21 positions and divide by reorderings within the 16 y positions and 5 x positions, giving coefficient 20349. Step identifiers and dependenciesRule: subset_count · Uses: G001, S033 S035 · impliesBefore C(21,16)=20349; choose y 16 times After 20349·x^5·y^16 Conditions: none The x exponent is 21-16=5; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S034 S036 · impliesBefore choose exactly 17 of 21 factors to contribute y After C(21,17)=51090942171709440000/(355687428096000·24)=5985 Conditions: none Count orderings of 21 positions and divide by reorderings within the 17 y positions and 4 x positions, giving coefficient 5985. Step identifiers and dependenciesRule: subset_count · Uses: G001, S035 S037 · impliesBefore C(21,17)=5985; choose y 17 times After 5985·x^4·y^17 Conditions: none The x exponent is 21-17=4; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S036 S038 · impliesBefore choose exactly 18 of 21 factors to contribute y After C(21,18)=51090942171709440000/(6402373705728000·6)=1330 Conditions: none Count orderings of 21 positions and divide by reorderings within the 18 y positions and 3 x positions, giving coefficient 1330. Step identifiers and dependenciesRule: subset_count · Uses: G001, S037 S039 · impliesBefore C(21,18)=1330; choose y 18 times After 1330·x^3·y^18 Conditions: none The x exponent is 21-18=3; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S038 S040 · impliesBefore choose exactly 19 of 21 factors to contribute y After C(21,19)=51090942171709440000/(121645100408832000·2)=210 Conditions: none Count orderings of 21 positions and divide by reorderings within the 19 y positions and 2 x positions, giving coefficient 210. Step identifiers and dependenciesRule: subset_count · Uses: G001, S039 S041 · impliesBefore C(21,19)=210; choose y 19 times After 210·x^2·y^19 Conditions: none The x exponent is 21-19=2; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S040 S042 · impliesBefore choose exactly 20 of 21 factors to contribute y After C(21,20)=51090942171709440000/(2432902008176640000·1)=21 Conditions: none Count orderings of 21 positions and divide by reorderings within the 20 y positions and 1 x positions, giving coefficient 21. Step identifiers and dependenciesRule: subset_count · Uses: G001, S041 S043 · impliesBefore C(21,20)=21; choose y 20 times After 21·x^1·y^20 Conditions: none The x exponent is 21-20=1; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S042 S044 · impliesBefore choose exactly 21 of 21 factors to contribute y After C(21,21)=51090942171709440000/(51090942171709440000·1)=1 Conditions: none Count orderings of 21 positions and divide by reorderings within the 21 y positions and 0 x positions, giving coefficient 1. Step identifiers and dependenciesRule: subset_count · Uses: G001, S043 S045 · impliesBefore C(21,21)=1; choose y 21 times After 1·x^0·y^21 Conditions: none The x exponent is 21-21=0; the two exponents still sum to 21. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S044 S046 · impliesBefore all k=0 through 21 have been accounted for After (x+y)^21=1·x^21·y^0 + 21·x^20·y^1 + 210·x^19·y^2 + 1330·x^18·y^3 + 5985·x^17·y^4 + 20349·x^16·y^5 + 54264·x^15·y^6 + 116280·x^14·y^7 + 203490·x^13·y^8 + 293930·x^12·y^9 + 352716·x^11·y^10 + 352716·x^10·y^11 + 293930·x^9·y^12 + 203490·x^8·y^13 + 116280·x^7·y^14 + 54264·x^6·y^15 + 20349·x^5·y^16 + 5985·x^4·y^17 + 1330·x^3·y^18 + 210·x^2·y^19 + 21·x^1·y^20 + 1·x^0·y^21 Conditions: none Every x/y choice belongs to exactly one k class, so this is the complete finite expansion, not just endpoint terms. Step identifiers and dependenciesRule: collect_all_terms · Uses: G001, S045 ## Reverse checks - S046 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained. - 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 (21, [[0, 1, 21, 0], [1, 21, 20, 1], [2, 210, 19, 2], [3, 1330, 18, 3], [4, 5985, 17, 4], [5, 20349, 16, 5], [6, 54264, 15, 6], [7, 116280, 14, 7], [8, 203490, 13, 8], [9, 293930, 12, 9], [10, 352716, 11, 10], [11, 352716, 10, 11], [12, 293930, 9, 12], [13, 203490, 8, 13], [14, 116280, 7, 14], [15, 54264, 6, 15], [16, 20349, 5, 16], [17, 5985, 4, 17], [18, 1330, 3, 18], [19, 210, 2, 19], [20, 21, 1, 20], [21, 1, 0, 21]], 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 38fcaa28ce86999a50dea4366babdfeee8010274dc252d7b8c10a72b2cb04b78 - candidate record SHA-256: 2be0c3d7aa548fde169c606b494e2f05d26e5bd4657bce042befab2e69984bd6 - semantic review receipt SHA-256: 4727d1bd957a9b7f6cadb7972ddb5ab80a31f5259dc5394c1dff2b71944ed36b - question-first witness SHA-256: 51e18649133238ac992229f6e16542a5f3af53ca73b2262ab0524508ea937dee - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261109-r05-binomial-theorem-0047.json)