What is the Number of Possible Nonequivalent Propositions with $P_1, P_2, P_3$ Using $\iff$ Operator?
You just need to use $P_1$, $P_2$, $P_3$ and $\Leftrightarrow$ symbols, for example $P_2$ is a formula, $(P_1 \Leftrightarrow P_3) \Leftrightarrow P_2$ is a formula. As stated in the question, you must not use $\vee$, $\wedge$ or $\neg$.
You need to find the number of non-equivalent formulae. For example, $P_1 \Leftrightarrow P_2$ is equivalent to $P_2\Leftrightarrow P_1$, but $P_1\Leftrightarrow P_2$ is not equivalent to $P_1 \Leftrightarrow P_3$.