The number of arrangements of six identical balls in three identical bins is ___________.

The value of the following limit is _____________.

$$\mathop {\lim }\limits_{x \to {0^ + }} {{\sqrt x } \over {1 - {e^{2\sqrt x }}}}$$

Which one of the following is the closed form for the generating function of the sequence (a_n}_{n $$\ge$$ 0} defined below?

$${a_n} = \left\{ {\matrix{ {n + 1,} & {n\,is\,odd} \cr {1,} & {otherwise} \cr } } \right.$$

Consider a simple undirected unweighted graph with at least three vertices. If A is the adjacency matrix of the graph, then the number of 3-cycles in the graph is given by the trace of