The $${2^n}$$ vertices of a graph $$G$$ correspond to all subsets of a set of size $$n$$, for $$n \ge 6$$. Two vertices of $$G$$ are adjacent if and only if the corresponding sets intersect in exactly two elements.

The maximum degree of a vertex in $$G$$ is

The $${2^n}$$ vertices of a graph $$G$$ correspond to all subsets of a set of size $$n$$, for $$n \ge 6$$. Two vertices of $$G$$ are adjacent if and only if the corresponding sets intersect in exactly two elements.

the number of vertices of degree zero in $$G$$ is

What is the cardinality of the set of integers $$X$$ defined below?
$$X = $$ {$$n\left| {1 \le n \le 123,\,\,\,\,\,n} \right.$$ is not divisible by either $$2, 3$$ or $$5$$ }

For each elements in a set of size $$2n$$, an unbiased coin in tossed. The $$2n$$ coin tosses are independent. An element is chhoosen if the corresponding coin toss were head.The probability that exactly $$n$$ elements are chosen is