In a three variable boolean function if all of the truth table rows are independent w.r.t the O/P, then total no of boolean function possible is $2^{2^3} = 256$
- because in the O/P of a truth table we have $2^n= 2^3$ places to fill with 0 or 1.
- This is possible because we have not specified any constraint on the boolean function.
The question here provide us with a constraint equation $f(x,y,z) = f(y',x',z)$
because of this constraint, all 8 rows of the truth table are not independent now. They are in 6 groups.
- $f(0,0,0) \ \ \ \text{ and } \ \ f(1,1,0)$ are having same O/P.
- $f(0,0,1) \text{ and } f(1,1,0)$ are having same O/P.
- $f(0,1,0) ,f(0,1,1) ,f(1,0,0) ,f(1,0,1) ,$ are independent.
So in these 6 rows of the truth table, we can fill up O/P with 0 or 1.
$\Rightarrow$ total $2^6$ functions possible.
note : we can also solve by calculating no of minterms and taking combinations of them.