If G is a finite group and H, K are subgroups of G, then how many H\G/K double cosets are there?
this is a good question! the only formula for the number of double cosets of $\displaystyle H, K$ i know is in terms of the conjugacy classes of $\displaystyle G,$ that is if $\displaystyle \mathcal{C}_j, \ 1 \leq j \leq m$ are the conjugacy classes of $\displaystyle G,$
then the number of double cosets of $\displaystyle H, K$ is equal to: $\displaystyle \ell=\frac{|G|}{|H||K|}\sum_{j=1}^m \frac{|\mathcal{C}_j \cap H||\mathcal{C}_j \cap K|}{|\mathcal{C}_j|}.$ you should check this formula for the special case when $\displaystyle H$ or $\displaystyle K$ is normal. note that if, for example
$\displaystyle K$ is normal, then $\displaystyle \forall g \in G: \ HgK=HKg.$ thus: $\displaystyle |HgK|=|HK|=\frac{|H||K|}{|H \cap K|}.$