How many valuations of these literals satisfy this expression? considering all the possible valuations of literals A, B, C, D, E, F, G and H (256 valuations in total), how would you go about finding how many of these valuations satisfy this expression:
$$ (A\rightarrow B) \wedge (B\rightarrow C) \wedge (D\rightarrow E) \wedge (F\rightarrow G) \wedge (G\rightarrow H)$$
I know I could draw the truth table with all the possible valuations, but what's a more efficient method? The solution that I calculated through software simulation should be 48.

Observe that the three groups $\{A,B,C\}$, $\{D,E\}$, and $\{F,G,H\}$ are essentially independent; we can analyze them separately.
In a group like $\{A, B, C\}$, not all truth values are possible, if $A$ is true then so are $B$ and $C$; if $B$ is true then so is $C$.  If we consider $A$ to be ‘earlier’ then $B$, and $B$ to be ‘earlier’ than $C$, then it suffices to know the earliest true value in the set of three; this completely determines the other values.  For example, if $B$ is the earliest true value than $A$ is false (since it is earlier than $B$) and $C$ is true (since $B\to C$).  So there there only four values this group can have: any of $A$, $B$, or $C$ can be the earliest true value, or else none is true; this is four possibilities.
$\{F,G,H\}$ similarly has four possible values.
$\{D,E\}$ similarly has three values: none is true; $E$ is the earliest true value, or $D$ is the earliest true value.
$4\cdot 4\cdot 3 = 48$.
