In the cluster expansion (section 5.2 in M. Kardar "Statistical Physics of Particles") we write the grand canonical partition function. During the expansion, we do the following switch between a sum and a product: $$ \sum\limits_{\{n_l\}} \prod_l \frac{1}{n_l!}\left(\frac{e^{l\beta\mu} b_l}{\lambda^{3l}l!}\right)^{n_l} = \prod_l \sum\limits_{n_l=0}^\infty \frac{1}{n_l!}\left(\frac{e^{l\beta\mu} b_l}{\lambda^{3l}l!}\right)^{n_l} $$ ($l$ is the size of the cluster, and $n_l$ is the number of clusters)

I'm trying to understand why this is fine. I figured that $ \sum\limits_{\{n_l\}} \prod_l$ means that we're going over all possible sets of cluster numbers (e.g 5 clusters of size 1, 3 of size 2, etc..) but in this case, what is the meaning of the product? How is the connection between $n_l$ and $l$ apparent?

$\prod_l \sum\limits_{n_l=0}^\infty $ is much clearer - for every cluster size, we look at all numbers of it possible.