How many different isomers of $C_3 H_4 Cl_2 F_2$ exist? I was on my high school chemistry class when I came across this problem, which, I think, belongs to group theory. The problem is that it is not possible to label the carbons as reflection should not be counted twice. (That is, 3,3-dichloro-2,2-difluoropropane is called 1,1-dichloro-2,2-difluoropropane)
(The carbon chain in the case of propane is fixed.)
And, how to solve the general problem of counting different isomers of 
$\require{mhchem}\ce{C_{n}H_{x}Cl_{y}F_{2n +2-x-y}}$ ?
Thank you. 
 A: $\rm C_3H_4Cl_2F_2$ is an alkane.   There are no multiple bonds or rings to worry about.   Further only one skeleton is permissible here: a three carbon chain.   The end two carbons have three groups attached, the middle carbon but two.   The carbon-carbon bonds can rotate freely so we only need worry about possible carbon chirality, and any mirror symmetry.
Noting that we have the same number of chlorine and fluorine atoms simplifies things as we can just check to see if exchanging them produces an isomer; but watch out for interaction with other symmetries.
$\begin{array}{ccc:ccc} &&& \text{middle C} & \text{end C} & X\leftrightarrow Y  \\ (\;)&(X)&(XY_2)& \times 2 & & \times 2\\ (\;)&(X_2)&(Y_2)&&&\times 2\\ (\;)&(XY)&(XY)&\times 2&\times 2&\div 2 & \text{the X-Y interchange cancels isometry}\\ (X)&(\;)&(XY_2)&&&\times 2\\ (X)&(X)&(Y_2)&\times 2&&\times 2\\ (X)&(Y)&(XY)&\times 2&\times 2&\times 2\\ (X)&(XY)&(Y)& \times 2\\ (X)&(Y_2)&(X)&&&\times 2 \\ (X_2)&(\;)&(Y_2)& \\ (XY)&(\;)&(XY) & \times 4&&\div 2 \end{array}$
I count 2729 isomers, but may well have missed others.
The general case gets much more complex.
A: Odd, I get 30 33 32 (last one is a meso isomer) stereoisomers from 16 constitutional isomers. Like so (red asterisks indicate a stereocenter) :

Sadly, the method to get this count was 'brute force enumeration' (in fact, a variant of the canonical path augmentation method of McKay for multigraphs). 
The stereocenters were deterined by eyeballing the structures and thinking about it - so they may well be wrong! 
Edit : Indeed, I missed the chiral centers on FC(Cl)CC(F)Cl. As to the more general question, I'm not sure how to 'extend' the series in some natural way.
For example, here are some counts of constitutional isomers for various similar formulae:
Formula|Count


*

*C4H5Cl3F2 86

*C4H6Cl2F2 59

*C5H6Cl3F3 632


All found using the same method.
