# Question 88: 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-0038) 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. P250-0088 · q-20261014-advance-20261109-r05-binomial-theorem-0038 ## Original question and givens Use natural exponent n=12, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary). - G001 — Use natural exponent n=12, 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=12≥0; x,y commute After (x+y)^12 is a finite product of 12 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 12 factors to contribute y After C(12,0)=479001600/(1·479001600)=1 Conditions: none Count orderings of 12 positions and divide by reorderings within the 0 y positions and 12 x positions, giving coefficient 1. Step identifiers and dependenciesRule: subset_count · Uses: G001, S001 S003 · impliesBefore C(12,0)=1; choose y 0 times After 1·x^12·y^0 Conditions: none The x exponent is 12-0=12; the two exponents still sum to 12. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S002 S004 · impliesBefore choose exactly 1 of 12 factors to contribute y After C(12,1)=479001600/(1·39916800)=12 Conditions: none Count orderings of 12 positions and divide by reorderings within the 1 y positions and 11 x positions, giving coefficient 12. Step identifiers and dependenciesRule: subset_count · Uses: G001, S003 S005 · impliesBefore C(12,1)=12; choose y 1 times After 12·x^11·y^1 Conditions: none The x exponent is 12-1=11; the two exponents still sum to 12. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S004 S006 · impliesBefore choose exactly 2 of 12 factors to contribute y After C(12,2)=479001600/(2·3628800)=66 Conditions: none Count orderings of 12 positions and divide by reorderings within the 2 y positions and 10 x positions, giving coefficient 66. Step identifiers and dependenciesRule: subset_count · Uses: G001, S005 S007 · impliesBefore C(12,2)=66; choose y 2 times After 66·x^10·y^2 Conditions: none The x exponent is 12-2=10; the two exponents still sum to 12. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S006 S008 · impliesBefore choose exactly 3 of 12 factors to contribute y After C(12,3)=479001600/(6·362880)=220 Conditions: none Count orderings of 12 positions and divide by reorderings within the 3 y positions and 9 x positions, giving coefficient 220. Step identifiers and dependenciesRule: subset_count · Uses: G001, S007 S009 · impliesBefore C(12,3)=220; choose y 3 times After 220·x^9·y^3 Conditions: none The x exponent is 12-3=9; the two exponents still sum to 12. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S008 S010 · impliesBefore choose exactly 4 of 12 factors to contribute y After C(12,4)=479001600/(24·40320)=495 Conditions: none Count orderings of 12 positions and divide by reorderings within the 4 y positions and 8 x positions, giving coefficient 495. Step identifiers and dependenciesRule: subset_count · Uses: G001, S009 S011 · impliesBefore C(12,4)=495; choose y 4 times After 495·x^8·y^4 Conditions: none The x exponent is 12-4=8; the two exponents still sum to 12. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S010 S012 · impliesBefore choose exactly 5 of 12 factors to contribute y After C(12,5)=479001600/(120·5040)=792 Conditions: none Count orderings of 12 positions and divide by reorderings within the 5 y positions and 7 x positions, giving coefficient 792. Step identifiers and dependenciesRule: subset_count · Uses: G001, S011 S013 · impliesBefore C(12,5)=792; choose y 5 times After 792·x^7·y^5 Conditions: none The x exponent is 12-5=7; the two exponents still sum to 12. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S012 S014 · impliesBefore choose exactly 6 of 12 factors to contribute y After C(12,6)=479001600/(720·720)=924 Conditions: none Count orderings of 12 positions and divide by reorderings within the 6 y positions and 6 x positions, giving coefficient 924. Step identifiers and dependenciesRule: subset_count · Uses: G001, S013 S015 · impliesBefore C(12,6)=924; choose y 6 times After 924·x^6·y^6 Conditions: none The x exponent is 12-6=6; the two exponents still sum to 12. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S014 S016 · impliesBefore choose exactly 7 of 12 factors to contribute y After C(12,7)=479001600/(5040·120)=792 Conditions: none Count orderings of 12 positions and divide by reorderings within the 7 y positions and 5 x positions, giving coefficient 792. Step identifiers and dependenciesRule: subset_count · Uses: G001, S015 S017 · impliesBefore C(12,7)=792; choose y 7 times After 792·x^5·y^7 Conditions: none The x exponent is 12-7=5; the two exponents still sum to 12. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S016 S018 · impliesBefore choose exactly 8 of 12 factors to contribute y After C(12,8)=479001600/(40320·24)=495 Conditions: none Count orderings of 12 positions and divide by reorderings within the 8 y positions and 4 x positions, giving coefficient 495. Step identifiers and dependenciesRule: subset_count · Uses: G001, S017 S019 · impliesBefore C(12,8)=495; choose y 8 times After 495·x^4·y^8 Conditions: none The x exponent is 12-8=4; the two exponents still sum to 12. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S018 S020 · impliesBefore choose exactly 9 of 12 factors to contribute y After C(12,9)=479001600/(362880·6)=220 Conditions: none Count orderings of 12 positions and divide by reorderings within the 9 y positions and 3 x positions, giving coefficient 220. Step identifiers and dependenciesRule: subset_count · Uses: G001, S019 S021 · impliesBefore C(12,9)=220; choose y 9 times After 220·x^3·y^9 Conditions: none The x exponent is 12-9=3; the two exponents still sum to 12. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S020 S022 · impliesBefore choose exactly 10 of 12 factors to contribute y After C(12,10)=479001600/(3628800·2)=66 Conditions: none Count orderings of 12 positions and divide by reorderings within the 10 y positions and 2 x positions, giving coefficient 66. Step identifiers and dependenciesRule: subset_count · Uses: G001, S021 S023 · impliesBefore C(12,10)=66; choose y 10 times After 66·x^2·y^10 Conditions: none The x exponent is 12-10=2; the two exponents still sum to 12. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S022 S024 · impliesBefore choose exactly 11 of 12 factors to contribute y After C(12,11)=479001600/(39916800·1)=12 Conditions: none Count orderings of 12 positions and divide by reorderings within the 11 y positions and 1 x positions, giving coefficient 12. Step identifiers and dependenciesRule: subset_count · Uses: G001, S023 S025 · impliesBefore C(12,11)=12; choose y 11 times After 12·x^1·y^11 Conditions: none The x exponent is 12-11=1; the two exponents still sum to 12. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S024 S026 · impliesBefore choose exactly 12 of 12 factors to contribute y After C(12,12)=479001600/(479001600·1)=1 Conditions: none Count orderings of 12 positions and divide by reorderings within the 12 y positions and 0 x positions, giving coefficient 1. Step identifiers and dependenciesRule: subset_count · Uses: G001, S025 S027 · impliesBefore C(12,12)=1; choose y 12 times After 1·x^0·y^12 Conditions: none The x exponent is 12-12=0; the two exponents still sum to 12. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S026 S028 · impliesBefore all k=0 through 12 have been accounted for After (x+y)^12=1·x^12·y^0 + 12·x^11·y^1 + 66·x^10·y^2 + 220·x^9·y^3 + 495·x^8·y^4 + 792·x^7·y^5 + 924·x^6·y^6 + 792·x^5·y^7 + 495·x^4·y^8 + 220·x^3·y^9 + 66·x^2·y^10 + 12·x^1·y^11 + 1·x^0·y^12 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, S027 ## Reverse checks - S028 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained. - 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 (12, [[0, 1, 12, 0], [1, 12, 11, 1], [2, 66, 10, 2], [3, 220, 9, 3], [4, 495, 8, 4], [5, 792, 7, 5], [6, 924, 6, 6], [7, 792, 5, 7], [8, 495, 4, 8], [9, 220, 3, 9], [10, 66, 2, 10], [11, 12, 1, 11], [12, 1, 0, 12]], 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: 93d2e4a077e414a3942bac16dce9e28863cb7ee99d86362e30633aa77ab7a49d - candidate record SHA-256: b39ff414a62b9e0342a5bf52a593ad5ec2bd1bb6ab42703ea58882e036d9220b - 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-0038.json)