- #1

Keen94

- 41

- 1

## 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-j}b

^{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!