Matrix multiplication is not commutative, so $P_0 \cdot P_1 \neq P_1 \cdot P_0$. I have a problem where I multiply $P_1^i \cdot P_0^{n-i}$, but not just in this way $P_1 \cdot P_1 \cdot \ldots \cdot P_0 \cdot \ldots \cdot P_0$, but all possible permutations of multiplying $i$ times the matrix $P_1$ with $n-i$ times the matrix $P_0$.
Is there a notation for this? Right now I'm writing $$ \Pi(P_1^i \cdot P_0^{n-i}) $$ and explain what I mean below.