Order of Double Coset I am working on a homework problem (so don't just give me the answer) from Herstein's Topics in Algebra, which goes as follows:

If $G$ is a finite group, show that the number of elements in the
  double coset $AxB$ is $$\dfrac{o(A)o(B)}{o(A\cap xBx^{-1})}$$

It makes sense to me, but my attempts at a proof seem to fall short of undeniability. I have been trying to show that $o(A \cap x B x^{-1})$ equals the number of duplicate terms in the list of all possible products of an element from $A$ and an element from $xB$. 
Suppose $a_0 \in A \cap xBx^{-1}$. Then $\exists b_0 \in B : a_0 = x b_0 x^{-1}$. In the list of products of the form $axb, a \in A, b \in B$, any term involving $a_0$ will be of the form $a_0 x b = x b_0 x^{-1} x b = x b_0 b$ for some $b \in B$. But then this list is just the left coset $xB$, which is already accounted for in the product list of $AxB$ by setting $a=e$. 
This is where I run into trouble. I can't seem to crystallize this argument. Am I going about this the right way? And if you see how I should extend this argument, can you nudge me in the right direction?
Thanks Math.SE.
 A: Edit: I didn't notice that "don't just give me the answer" in the first version of my answer. The following are some hints.
Firstly, the formula 
$$o(HK)=\frac{o(H)o(K)}{o(H\cap K)}$$
holds for subgroups $H$ and $K$ of $G$, but may not hold for subsets $H$ and $K$ of $G$.
Secondly, note that
$$o(AxB)=o(AxBx^{-1}).$$
A: This is a very nice formula:
$$|HK|=\frac{|H||K|}{|H\cap K|}.$$
There are a couple of beautiful proofs, suggested intrinsically by its form, that I will hint at:


*

*Coset spaces. You probably know the formula $|G/H|=|G|/|H|$; in other words you know how to interpret a ratio of two group orders as the size of a coset space when the denominator group is contained in the numerator group. There are four quantities appearing in the desired formula, arranged in the form $A=BC/D$. Can you rearrange this so that the equation involves expressions of the form $|X|/|Y|$? Now interpret the expressions as spaces of either left or right cosets. Exhibit an explicit bijection between these two coset spaces.

*Group actions. If $G$ acts on a set $X$, then the orbit-stabilizer formula asserts that the size of an orbit $Gx$ is equal to the index of $x$'s stabilizer in $G$. You know $|H||K|=|H\times K|$. Can you think of a group action on a set which yields the desired formula from orbit-stabilizer?

A: I am reading "Topics in Algebra 2nd Edition" by I. N. Herstein.
This problem is Problem 40 on p.49 in Herstein's book.
I solved this problem as follows:

$\#AxB=o(A(xBx^{-1}))=\frac{o(A)o(xBx^{-1})}{o(A\cap xBx^{-1})}=\frac{o(A)o(B)}{o(A\cap xBx^{-1})}$ by THEOREM 2.5.1 on p.45 in Herstein's book.
Note that $xBx^{-1}$ is a subgroup of $G$ for any $x\in G$ and $o(B)=o(xBx^{-1})$ for any $x\in G$.
Note that $G\ni y\mapsto yx^{-1}\in G$ is injective, so $\#AxB=o(A(xBx^{-1}))$.

