you need to identify the values of triplet (a,b,c) from $2^3$ possible values for which $a_1=c_2$ and $a_2=c_1$ for any two triplets $(a_1,b,c_1)$ and $(a_2,b,c_2)$ because for those values, $f(.)$ will be same according to the question. For example, $f(0,0,1) = f(1,0,0).$
So, out of $2^3$ values, the possible cases are:
$1)$ $\{(0,0,1),(1,0,0)\}$
$2)$ $\{(0,1,1),(1,1,0)\}$
$3)$ $(0,0,0)$
$4)$ $(0,1,0)$
$5)$ $(1,0,1)$
$6)$ $(1,1,1)$
Now, for each $6$ cases, you can assign either $0$ or $1$ as a value of the function and so, total possible boolean functions = $2^6$