The minimum number of colors required to color the vertices of a cycle with $$n$$ nodes in such a way that no two adjacent nodes have the same colour is:

The minimum number of cards to be dealt from an arbitrarily shuffled deck of 52 cards to guarantee that three cards are from same suit is

The solution to the recurrence equation
$$T\left( {{2^k}} \right)$$ $$ = 3T\left( {{2^{k - 1}}} \right) + 1$$,
$$T\left( 1 \right) = 1$$ is:

The number of binary strings of $$n$$ zeros and $$k$$ ones such that no two ones are adjacent is: