Consider the following Boolean expression for $$F:$$
$$F\left( {P,\,Q,\,R,\,S} \right) = PQ + \overline P QR + \overline P Q\overline R S.$$
The minimal sum-of-products form of $$F$$ is

The maximum number of edges in a bipartite graph on $$12$$ vertices is _________.

Suppose you break a stick of unit length at a point chosen uniformaly at random. Then the expected length of the shorter stick is __________________.

Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is X/1296. The value of X is__________