Calculating the number of assignments satisfying a general propositional formula

Question

I know, for a disjunctive clause of the form $x_1 \vee ... \vee x_i$, the number of assignments satisfying it is simply $2^i - 1$, but what about for a general formula? Is the number of satisfying assignments be calculated polynomially?

In Papadimitriou's Computational Complexity (p. 301), when explaining an approximation algorithm for $k$-MAXGSAT where the input is, over $n$ variables, $\Phi = \{\phi_1, ..., \phi_n\}$ with $\phi_i$ being any general formula involving at most $k$ variables, it is stated that

... Each expression $\phi_i \in \Phi$ involves $k$ Boolean variables. Out of the $2^k$ truth assignments, we can easily calculate the number $t_i$ of truth assignments that satisfy $\phi_i$. ...

But I don't find it obvious how one can compute it easily, as even with Tseitin's method, the transformed result of $\phi_i$ wouldn't necessarily be a single disjunctive clause. Where did I go wrong?