The average score for a n-dimensional convex polytope collection is?

A $n$ -dimensional convex polytope has counts of each entity class: a null set (entity ${}_{n,-1}$ ) [always just 1], vertices (entity ${}_{n,0}$ ), edges (entity ${}_{n,0}$ ), faces (entity ${}_{n,2}$ ), etc. up to (entity ${}_{n,n}$ ) [the object itself and again just 1]. So that words do not have to be invented for each (entity ${}_{n,m}$ -dimensional entity type. I will be referring to them as entity ${}_{n,m}$ or entities ${}_{n\,m}$ as appropriate. Please, note that the full dimension entity and the null set entity are also counted.

Given some random collection of convex polytopes embedded in an appropriate genus 0 space (that is, it has no holes) of any integer dimensionality greater or equal to 0 of cardinality greater than 0 (in other words, there is at least one convex polytope and all convex polytopes have at least a vertex), what is the average score of the collection?

An individual convex polytope's score, whose index is $m$ , is $\sum _{j=-1}^n ((-1)^j \text{entity}_{m,j})$ .