# A full binomial expansion by choosing y factors

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This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented.

derivation-mko-binomial-theorem-01 · ocme-new142-reader-adoption-20261021



## Goal and scope

Establish coefficient counting and expand n=4 completely.

Nonnegative integer n and commuting x,y; no noninteger exponents.





## Givens and symbols


- work


- G1 — n belongs to N0 and x,y are in a commutative real polynomial ring; the instance is n=4.


- k — number of y-factors
- S — positions choosing y





## Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.


- mko_definition · cited — For n∈N_0, (x+y)^n=sum over k=0..n of C(n,k)*x^(n−k)*y^k. Coefficients are exact, all indices and both endpoint terms are retained, and the exponents must not be interchanged. At n=0 the polynomial is constant 1. A combinatorial witness chooses k y-factors from n factors. Negative/noninteger exponents and noncommuting variables are outside this finite-polynomial slice. · mko-binomial-theorem
- work · prerequisite — Finite distributivity, commutative multiplication and the count comb(n,k)=n!/[k!(n-k)!] of size-k subsets; the empty product is 1.





## Main derivation

S01 · impliesBefore

(x+y)^n=\prod_{i=1}^{n}(x+y)

After

\sum_{S\subseteq\{i\in\mathbb Z:1\le i\le n\}}x^{n-|S|}y^{|S|}

Conditions: none

Rule: work · Uses: G1, work

Leaf basis: work

Choose x or y in each factor, indexing the full distributive expansion by positions S choosing y.



S02 · impliesBefore

|S|=k

After

\#\{S:|S|=k\}=\binom nk,\quad x^{n-|S|}y^{|S|}=x^{n-k}y^k

Conditions: none

Rule: work · Uses: G1, S01, work

Leaf basis: work

Group by the number k of y-factors; each group has comb(n,k) terms.



S03 · impliesBefore

k\in\{j\in\mathbb Z:0\le j\le n\}

After

(x+y)^n=\sum_{k=0}^{n}\binom nk x^{n-k}y^k

Conditions: none

Rule: work · Uses: G1, S02, work

Leaf basis: work

Subset sizes partition every choice, including endpoints k=0 and k=n.



S04 · impliesBefore

n=4,\quad k=0,1,2,3,4

After

\binom40,\binom41,\binom42,\binom43,\binom44=1,4,6,4,1

Conditions: none

Rule: work · Uses: G1, S03, work

Leaf basis: work

Use the factorial combination formula for all five coefficients.



S05 · impliesBefore

(x+y)^4

After

x^4+4x^3y+6x^2y^2+4xy^3+y^4

Conditions: none

Rule: work · Uses: G1, S04, work

Leaf basis: work

Pair every coefficient with the correct exponents.



S06 · impliesBefore

x=2,\quad y=-1

After

(2-1)^4=16-32+24-8+1=1

Conditions: none

Rule: work · Uses: G1, S05, work

Leaf basis: work

An exact evaluation at negative y checks signs and endpoint terms.



S07 · impliesBefore

n=0

After

(x+y)^0=1=\binom00x^0y^0

Conditions: none

Rule: work · Uses: G1, S06, work

Leaf basis: work

The empty product and sole empty subset give constant 1.



S08 · impliesBefore

xy\ne yx\quad\text{(outside scope)}

After

(x+y)^2=x^2+xy+yx+y^2\text{ need not equal }x^2+2xy+y^2

Conditions: none

Rule: work · Uses: G1, S07, work

Leaf basis: work

Display the noncommuting boundary, identifying the assumption needed for grouping.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S08 · premise_audit · G1, S07, work — Do not extend the commutative proof to arbitrary matrices.
- S07 · premise_audit · G1, S06, work — This is a polynomial identity with the constant-polynomial convention even at x+y=0.
- S06 · premise_audit · G1, S05, work — Odd y-powers are negative and even powers positive.
- S05 · premise_audit · G1, S04, work — Each monomial has total degree4.
- S04 · premise_audit · G1, S03, work — The middle coefficient is 6, not 4.
- S03 · premise_audit · G1, S02, work — The y-exponent is the selected count k; roles are not exchanged.
- S02 · premise_audit · G1, S01, work — Commutativity permits choices at different positions to merge into one monomial.
- S01 · premise_audit · G1, work — The index set contains every integer1≤i≤n; no factor choices are omitted.





## Error controls and negative controls

Exponentsk y-choices correspond to x^(n-k)y^k.

All coefficientsKeep all five terms including endpoints; the exact polynomial grid checks the n=4 identity.


- (x+y)^4 is only x4+y4. → reject — At x=y=1 the left side is 16 and the right2.
- n=-1 has the same finite expansion. → reject — The scope requires n in N0; a negative power is not this polynomial.
- Arbitrary matrices permit merging xy and yx. → reject — For noncommuting matrices they can differ.





## Result and limitations

Grouping all subsets proves the binomial formula; at n=4 coefficients are 1,4,6,4,1.







## Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: ee47015e5d6cac87bd69650c5b71ea2b0a20f9578bdc2c3e785bdccb17535ada
- original source SHA-256: a2fc26decd1525ce395cef607d62fc498d1cac913a33f9d6de68ffd0ffe794d8
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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