Math 2.0

Combinatorics Proof

Proof of the alternating sum of binomial coefficients from the binomial theorem

Builds on Binomial theoremBinomial coefficient

Proof

  1. Let n≥1n \ge 1. The integers ℤ\Z form a commutative ring. Apply the binomial theorem in ℤ\Z with x=−1x = -1 and y=1y = 1.
  2. It gives (−1+1)n=∑k=0n(nk)(−1)k1n−k=∑k=0n(−1)k(nk)(-1 + 1)^n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k 1^{n-k} = \sum_{k=0}^{n} (-1)^k \binom{n}{k}.
  3. The left side is 0n0^n. Since n≥1n \ge 1, we have 0n=0⋅0n−1=00^n = 0 \cdot 0^{n-1} = 0.
  4. Hence ∑k=0n(−1)k(nk)=0\sum_{k=0}^{n} (-1)^k \binom{n}{k} = 0.

Idea

Putting x=−1x = -1 and y=1y = 1 in the binomial theorem gives each coefficient the sign (−1)k(-1)^k. The left side becomes 0n0^n, which is 00 as soon as n≥1n \ge 1.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first. The path takes the proofs Proof of Pascal's rule from the recursive definition, Proof of the factorial formula for binomial coefficients by induction, Proof of the binomial theorem by induction.

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Binomial coefficient, definition, no evidence levelBinomial coefficientdefinitionFactorial, definition, no evidence levelFactorialdefinitionGroup, concept, no evidence levelGroupconceptProof of Pascal's rule from the recursive definition, proof, no evidence levelProof of Pascal's rul…proofRing, definition, no evidence levelRingdefinitionPascal's rule, statement, no evidence levelPascal's rulestatementCommutative ring, definition, no evidence levelCommutative ringdefinitionProof of the factorial formula for binomial coefficients by induction, proof, no evidence levelProof of the factoria…proofFactorial formula for binomial coefficients, statement, no evidence levelFactorial formula for…statementProof of the binomial theorem by induction, proof, no evidence levelProof of the binomial…proofBinomial theorem, statement, no evidence levelBinomial theoremstatementProof of the alternating sum of binomial coefficients from the binomial theorem, proof, no evidence levelProof of the alternat…proof
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Binomial coefficient definition
  2. Factorial definition
  3. Group concept
  4. Proof of Pascal's rule from the recursive definition proof
  5. Ring definition
  6. Pascal's rule statement
  7. Commutative ring definition
  8. Proof of the factorial formula for binomial coefficients by induction proof
  9. Factorial formula for binomial coefficients statement
  10. Proof of the binomial theorem by induction proof
  11. Binomial theorem statement
  12. Proof of the alternating sum of binomial coefficients from the binomial theorem proof

Where this leads

7 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Alternating_Sum_and_Difference_of_Binomial_Coefficients_for_Given_n
About this page
TagGYM930WM7E
URIm2://GYM930WM7E
Kindproof
WritingW0 stub
MSC05A10
ProvesAlternating sum of binomial coefficients
UsesBinomial theorem, Binomial coefficient
Changed4 November 2026

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