Math 2.0

Combinatorics Proof

Proof of the sum of binomial coefficients from the binomial theorem

Builds on Binomial theoremBinomial coefficient

Proof

  1. The integers ℤ\Z form a commutative ring. Apply the binomial theorem in ℤ\Z with x=y=1x = y = 1.
  2. It gives (1+1)n=∑k=0n(nk)1k1n−k(1 + 1)^n = \sum_{k=0}^{n} \binom{n}{k} 1^k 1^{n-k}.
  3. Every power of 11 equals 11, so the right side is ∑k=0n(nk)\sum_{k=0}^{n} \binom{n}{k}.
  4. The left side is 2n2^n. Hence ∑k=0n(nk)=2n\sum_{k=0}^{n} \binom{n}{k} = 2^n.

Idea

The binomial theorem with both variables equal to 11 removes the monomials and leaves only the coefficients. Their sum is then (1+1)n=2n(1 + 1)^n = 2^n.

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 sum of binomial coefficients from the binomial theorem, proof, no evidence levelProof of the sum of b…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 sum of binomial coefficients from the binomial theorem proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Sum_of_Binomial_Coefficients_over_Lower_Index
About this page
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URIm2://GS2SF16HQY
Kindproof
WritingW0 stub
MSC05A10
ProvesSum of binomial coefficients
UsesBinomial theorem, Binomial coefficient
Changed4 November 2026

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