What is the maximum number of different Boolean functions involving $n$ Boolean variables?
We have n boolean variables
So, we will have total of 2n combination of truth table values
For each of these 2n values, to define a boolean function they may be 0 or 1.
So we have 2 choices each for each 2n combination of truth table values
Hence the total number of boolean functions possible with n variables is $2^{2^{n}}$
With n boolean variables total number of rows in the truth table = 2^n
Each row of truth table can be taken or not taken (only 2 choices) .
So, maximum no of different boolean functions possible = 2^(2^n)
The number of m-ary functions in p-valued algebra having n-variables is given by $m^{p^{n}}$ Here, m = 2 (boolean functions), p = 2 (boolean variables) and n = n (number of variables) So, total functions = $2^{2^{n}}$ Alternative approach : With n boolean variables, we can have ${2^{n}}$ combinations for functions. And now since each of these is designed to be a boolean function, it's output value can be either 1 or 0, i.e. 2 choices for each function. So total such functions = $2^{2^{n}}$