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An $n-$variable Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ is called symmetric if its value depends only on the number of $1’s$ in the input. Let $\sigma_n$ denote the number of such functions.

1. Calculate the value of $\sigma_4$.
2. Derive an expression for $\sigma_n$ in terms of $n$.

Since, the value of the function is dependent only on the number of 1's in the n-bit string .
The possible number of ones are : 0,1,2......,n. There are n+1 possibilities.
Now map each of these possibilities of number of ones to 0 or 1.

Therefore no. of such functions = $2^{n+1}$.

$\sigma_4$=$2^{4+1}$=32