2n+1C0−2n+1C1+...+2n+1C2n=

1+x2n+1=2n+1C0+2n+1C1x+2n+1C2x2+...+2n+1C2nx2n+2n+1C2n+1.x2n+1For x=-1, we have 0=C0   2n+1-C1   2n+1+C2   2n+1-........+C2n   2n+1-C2n+1   2n+1 ⇒C0   2n+1-C1   2n+1+C2   2n+1-........+C2n   2n+1=C2n+1   2n+1=1