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A boolean function on $n$ variables is a function $f$ that takes an n-tuple of boolean values $x \in \{0,1\}^n$ as input and produces a boolean value $f(x)\in \{0,1\}$ as output.  

We say that a boolean function $f$ is symmetric if, for all inputs $x,y \in \{0,1\}^n$ with the same number of zeros (and hence the same number of ones), $f(x)=f(y)$. What is the number of symmetric boolean functions on $n$ variables?

  1. $n+1$
  2. $n!$
  3. $\displaystyle \sum^n_{i=0} \begin{pmatrix} n\\i \end{pmatrix}$ 
  4. $2^{n+1}$
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