Note: NOT is absent in function $f$.
for two boolean variables, $p_{1}=1$,$p_{2}=1$, neither $p_{1}\wedge p_{2}$ nor $p_{1} \vee p_{2}$ is $0$. ie, $f(1,1)$ is never $0.$
for $i = 1$ to $n$. $f(p_{i})$ is a function of AND,OR operations on $p_{i}$. if all $p_{i}=1$, then $f$ can never be $0$;
similarly, if all $p_{i}=0,$ $f$ can never be $1$;
Therefore ${A,B}$ are not possible.
for $j<n$, if all $p_{j}=0$ and $p_{n-j}=1$,then $f(p_{j}, p_{n-j}) =$majority if each $0$ is AND with each $1.$The remaining $1's$ or $0's$ are OR with the result.
Hence, MAJORITY can be computed from $f$.
Option C is possible.
To check odd number of $1's$, for PARITY function, we have to get the result $0$ for even number of $1's$ which is not possible with just AND and OR operations, how might we combine(since NOT is absent in $f$);
D is not possible.
For option E, we check by symmetry. When the inputs are complemented among $0's$ and $1's$, can $f$ change to $f'$? $f$ is not always fixed for a particular input,. example, $f(0,1) = 0\vee 1=1$ $0\wedge 1=0$,hence $f$ can take multiple values for same input. Therefore E is also not right.
The only possible answer is C .'Hence A,B,D,E are not possible.