# Question 78: 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-0028) 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. P250-0078 · q-20261014-advance-20261109-r05-binomial-theorem-0028 ## Original question and givens Use natural exponent n=2, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary). - G001 — Use natural exponent n=2, 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=2≥0; x,y commute After (x+y)^2 is a finite product of 2 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 2 factors to contribute y After C(2,0)=2/(1·2)=1 Conditions: none Count orderings of 2 positions and divide by reorderings within the 0 y positions and 2 x positions, giving coefficient 1. Step identifiers and dependenciesRule: subset_count · Uses: G001, S001 S003 · impliesBefore C(2,0)=1; choose y 0 times After 1·x^2·y^0 Conditions: none The x exponent is 2-0=2; the two exponents still sum to 2. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S002 S004 · impliesBefore choose exactly 1 of 2 factors to contribute y After C(2,1)=2/(1·1)=2 Conditions: none Count orderings of 2 positions and divide by reorderings within the 1 y positions and 1 x positions, giving coefficient 2. Step identifiers and dependenciesRule: subset_count · Uses: G001, S003 S005 · impliesBefore C(2,1)=2; choose y 1 times After 2·x^1·y^1 Conditions: none The x exponent is 2-1=1; the two exponents still sum to 2. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S004 S006 · impliesBefore choose exactly 2 of 2 factors to contribute y After C(2,2)=2/(2·1)=1 Conditions: none Count orderings of 2 positions and divide by reorderings within the 2 y positions and 0 x positions, giving coefficient 1. Step identifiers and dependenciesRule: subset_count · Uses: G001, S005 S007 · impliesBefore C(2,2)=1; choose y 2 times After 1·x^0·y^2 Conditions: none The x exponent is 2-2=0; the two exponents still sum to 2. Step identifiers and dependenciesRule: group_like_terms · Uses: G001, S006 S008 · impliesBefore all k=0 through 2 have been accounted for After (x+y)^2=1·x^2·y^0 + 2·x^1·y^1 + 1·x^0·y^2 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, S007 ## Reverse checks - S008 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained. - 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 (2, [[0, 1, 2, 0], [1, 2, 1, 1], [2, 1, 0, 2]], 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: c00566b36326b791c4ab9313a94694fafa92b2e45d3b78f0935fd7f56275ac21 - candidate record SHA-256: 18bf64a4f6549972942da6ab85c14b27d6c8993fbc516b95f008dc7f7725b8c0 - 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-0028.json)