$$n$$ couples are invited to a party with the condition that every husband should be accompanied by his wife. However, a wife need not be accompanied by her husband. The number of different gatherings possible at the party is

Let $$A$$ be a sequence of $$8$$ distinct integers sorted in ascending order. How many distinct pairs of sequence, $$B$$ and $$C$$ are there such that
i) Each is sorted in ascending order.
ii) $$B$$ has $$5$$ and $$C$$ has $$3$$ elements, and
iii) The result of merging $$B$$ $$C$$ gives $$A$$?

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