Let $P\in M_{k\times n}$ and $A\in M_{n\times k}$ then $$det(PA)=\sum_{\sigma\in S_k}\text{sgn}(\sigma)\prod_i (PA)_{i,\sigma(i)}$$ $$=\sum_{\sigma\in S_k}\text{sgn}(\sigma)\prod_i \sum_jP_{ij}A_{j,\sigma(i)}$$ $$\overset{(*)}{=}\sum_{j_1,...,j_k}P_{1,j_1}\cdots P_{k,j_k}(\sum_\sigma \text{sgn}(\sigma)\cdot A_{j_1,\sigma(1)}\cdots A_{j_k,\sigma(k)})$$
I don't understand the last transition $(*)$, this was taken as part of some proof in ring theory. $j_1,\dots,j_n$ weren't defined before $(*)$ so you just kind of need to figure out what are they. Can someone help me understand that transition? Thanks.