Isomers of Propionaldehyde

Question 12.1. How many chlorinated isomers has propionaldehyde We write 

We indicate now the formalism, how we can compute the number of isomers We identify the a-C atom, as a box, and the (1-C atom likewise as a box. In the box, which refers to the a-C atom, now two hydrogen atoms, hydrogen atom and one chlorine atom, or two chlorine atoms can be. In the box, which refers to the fi-C atom, either three hydrogen atoms can be, or one hydrogen atom and two chlorine atoms, or two hydrogen atoms and one chlorine atom, or three chlorine atoms. 

For the a-C atom, there is one possibility to have zero chlorine atoms attached, one possibility to have one chlorine atom attached, and one possibility to have two chlorine atoms attached. So, each coefficient in the generating function is one and we get for the a-C atom 1 + x + x. We repeat the same consideration for the j8-C atom and obtain 1 + x + x + x as the generating function. 

The generating function for the entire molecule is simply the product of these two generating functions  

The coefficients at that kth powers of x refer to the amount of that isomers with k chlorine atoms. If we want to know the total number of isomers from 0 Cl to 5 Cl, then we put x = 1, c.f., also Fig. 12.1. The procedure is not restricted to chlorinated isomers, of course. 

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