Let $G$ be a group and let $S_1, \dots, S_k$ be subgroups. Consider the product of the $\{S_i\}$:
$$S_1 S_2 \dots S_k := \{s_1 s_2 \dots s_k \ \mid \ s_i \in S_i \text{ for each } i\}$$
It's well known that $$|S_1 S_2| = |S_1||S_2| / |S_1 \cap S_2|.$$
Is there a similar formula to compute $|S_1 S_2 \dots S_k|$?
Note that $G$ is possibly non-abelian and so the product of subgroups is not necessarily itself a subgroup, so we can't just apply the above formula recursively.