There are three cities A, B and C. Each of these cities is connected with the other two cities by at least one direct road. If a traveller wants to go from on city (origin) to another city (destination), she can do so either by traversing a road connecting the two cities directly, or by traversing two roads, the first connecting the origin to the third city and the second connecting the third city to the destination. In all there are 33 routes from A to B (including those via C). Similarly, there are 23 routes from B to C (including those via A). How many roads are there from A to C directly?

In how many ways is it possible to choose a white square and a black square on a chess board so that the squares must not lie in the same row or column?

In how many distinguishably different ways can a cube be painted using at most two colours - White and Black - such that each face is coloured with exactly one of the two given colours?

Two different teams with four students each were training for an international math contest. The students were given an assessment exam and their scores were ranked. What is the probability that the four best scores come from only one of the teams assuming that no two students got the same score?