Para encontrarmos o valor de P, realizaremos os cálculos abaixo:

\(\begin{align} & AP+BP=3PC \\ & (P-A)+(P-B)=3(C-P) \\ & (x-1,y+1,z)+(x-1,y,z-1)=3(0-x,1-y,2-z) \\ & (2x-2,2y+1,2z-1)=(-3x,3-3y,6-3z) \\ & \\ & 2x-2=-3x \\ & x=\frac{2}{5} \\ & \\ & 2y+1=3-3y \\ & y=\frac{2}{5} \\ & \\ & 2z-1=6-3z \\ & z=\frac{7}{5} \\ & \\ & P=\left( \frac{2}{5},\frac{2}{5},\frac{7}{5} \right) \\ \end{align} \)

Portanto, o ponto P será \(\boxed{P = \left( {\frac{2}{5},\frac{2}{5},\frac{7}{5}} \right)}\).