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Proofs using the binomial theorem

  1. Jun 10, 2015 #1
    1. The problem statement, all variables and given/known data
    Prove that nj=0(-1)j(nCj)=0


    2. Relevant equations
    Definition of binomial theorem.

    3. The attempt at a solution
    If n∈ℕ and 0≤ j < n then 0=nj=0(-1)j(nCj)
    We know that if a,b∈ℝ and n∈ℕ then (a+b)n=∑nj=0(nCj)(an-jbj)

    Let a=1 and b= -1 so that 0=(1+(-1))n=∑nj=0(nCj)(1n-j(-1)j)

    LHS=∑nj=0(nCj)(1)(-1)j) since (1n-j)=+1

    LHS=∑nj=0(-1)j(nCj)

    Is this the best way to prove it or is the induction buisness better? Thanks in advance!
     
  2. jcsd
  3. Jun 10, 2015 #2
    that's how I would have done it
     
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