The question is asking – “How many compound propositions are there that are not logically equivalent to each other?”
With 4 variables, we can have $2^4$ = 16 rows in the truth table.
Now, these 16 rows can be filled in 2^16 = 65536 different ways (as each row has only 2 possibilities i.e., either T or F, so using multiplication rule, total possibilities = 2x2x2x…..x2(16 times) = 2^16 = 65536)
Now, corresponding to each way of filling the truth table, we have a particular compound proposition that is different from the compound proposition obtained from the filling of truth table in a different way.
Since, there are 65536 different ways of filling the truth table, we can have 65536 different compound propositions that are NOT logically equivalent. Hence, answer is 65536.