Hello,

Whew ! I finally spotted the mistake !

You're taking z>0, while, since the plane x+y+z=1 intersects the xy-plane, there are some negative z that satisfy the conditions !

Actually, we have z=1-x-y>1-2=-1, because x+y<2 (large inequalities, but I'm too lazy to do the latex)

So try it with the boundaries for z : -1 and 1-x-y

(I haven't done the calculations yet)