Proofs using the binomial theorem

[Keen94]

Homework Statement

Prove that ∑ⁿ_j=0(-1)^j(nCj)=0

Homework Equations

Definition of binomial theorem.

The Attempt at a Solution

If n∈ℕ and 0≤ j < n then 0=∑ⁿ_j=0(-1)^j(nCj)
We know that if a,b∈ℝ and n∈ℕ then (a+b)ⁿ=∑ⁿ_j=0(nCj)(a^n-jb^j)

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

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

LHS=∑ⁿ_j=0(-1)^j(nCj)

Is this the best way to prove it or is the induction business better? Thanks in advance!

 

[fourier jr]

that's how I would have done it