Let $$G$$ be an arbitrary graph with $$n$$ nodes and $$k$$ components. If a vertex is removed from $$G$$, the number of components in the resultant graph must necessarily lie between

$$m$$ identical balls are to be placed in $$n$$ distinct bags. You are given that $$m \ge kn$$, where $$k$$ is a natural number $$ \ge 1$$. In how many ways can the balls be placed in the bags if each bag must contain at least $$k$$ balls?

$$n$$ 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 $$A$$?