Let G(V, E) be a directed graph, where V = {1, 2, 3, 4, 5} is the set of vertices and E is the set of directed edges, as defined by the following adjacency matrix A.

$$A[i][j] = \left\{ {\matrix{ {1,} & {1 \le j \le i \le 5} \cr {0,} & {otherwise} \cr } } \right.$$

A[i][j] = 1 indicates a directed edge from node i to node j. A directed spanning tree of G, rooted at r $$\in$$ V, is defined as a subgraph T of G such that the undirected version of T is a tree, and T contains a directed path from r to every other vertex in V. The number of such directed spanning trees rooted at vertex 5 is _____________.

Consider the following directed graph: 

Which of the following is/are correct about the graph?

Define R_n to be the maximum amount earned by cutting a rod of length n meters into one or more pieces of integer length and selling them. For i > 0, let p[i] denotes the selling price of a rod whose length is i meters. Consider the array of prices:

p[1] = 1, p[2] = 5, p[3] = 8, p[4] = 9, p[5] = 10, p[6] = 17, p[7] = 18

Which of the following statements is/are correct about R₇?

Assume that multiplying a matrix $${G_1}$$ of dimension $$p \times q$$ with another matrix $${G_2}$$ of dimension $$q \times r$$ requires $$pqr$$ scalar multiplications. Computing the product of $$n$$ matrices $${G_1}{G_2}{G_{3...}}{G_n}$$ can be done by parenthesizing in different ways. Define $${G_i}\,\,{G_{i + 1}}$$ as an explicitly computed pair for a given paranthesization if they are directly multiplied. For example, in the matrix multiplication chain $${G_1}{G_2}{G_3}{G_4}{G_5}{G_6}$$ using parenthesization $$\left( {{G_1}\left( {{G_2}{G_3}} \right)} \right)\left( {{G_4}\left( {{G_5}{G_6}} \right)} \right),\,\,{G_2}{G_3}$$ and $${G_5}{G_6}$$ are the only explicitly computed pairs.

Consider a matrix multiplication chain $${F_1}{F_2}{F_3}{F_4}{F_5},$$ where matrices $${F_1},{F_2},{F_3},{F_4}$$ and $${F_5}$$ are of dimensions $$2 \times 25,\,\,25 \times 3,\,\,3 \times 16,\,\,16 \times 1$$ and $$1 \times 1000,$$ respectively. In the parenthesization of $${F_1}{F_2}{F_3}{F_4}{F_5}$$ that minimizes the total number of scalar multiplications, the explicitly computed pairs is/are