Calculating a sum that's taken over all of the k-element subspaces of an n-element space

Currently I constructed a formula (the idea from which it arose was completely combinatorial, won’t bother you with that) and was wondering whether this is applic-able in Mathematica (I’ve used it so far, primarily for implementation of Numerical Methods and visualisation of ODEs). The idea is as follows:

Given $n in mathbb N$ $setminus ${0,1} and fixing $k in n setminus${$emptyset$} and for each $i in {overline{ 1,n}}$ I’m given the following numbers (they are known) – $L_{i},I_{i}$ and $m_{i}$. And now I want to calculate the following sum (a somewhat "similar" sum, meaning that it’s taken over subsets, arises in the Inclusion-Exclusion principle):

$$sum_{Jsubseteq lbrace 1,…,n rbrace Card(J)=k} prod_{i in Jj in lbrace1,…,n rbrace setminus J}(I_j-J_jm_j)L_im_i$$

$Card(J)$ clearly stands for the cardinality of the specific taken subset. It’s clear that the above expression will consist of $binom{n}{k}$ amount of additions, every one of each will consist of $n$ multipliers. This is the formula. Thank you in advance.

p.s. I can construct an Inclusion-Exclusion alike formulation of the above stated sum, I’m asking out of curiosity.