Balancedness and correlation immunity of symmetric Boolean functions

New subsets of symmetric balanced and symmetric correlation immune functions are identified. The method involves interesting relations on binomial coefficients and highlights the combinatorial richness of these classes. As a consequence of our constructive techniques, we improve upon the existing lower bounds on the cardinality of the above sets. We consider higher order correlation immune functions and show how to construct n  -variable, third order correlation immune function for each perfect square n⩾9n⩾9.