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

The number of possible min-heaps containing each value from $$\left\{ {1,2,3,4,5,6,7} \right\}$$ exactly once is _____.

Consider the weights and values of items listed below. Note that there is only one unit of each item.

Item number	Weight (in Kgs)	Value (in Rupees)
1	10	60
2	7	28
3	4	20
4	2	24

The task is to pick a subset of these items such that their total weight is no more than $$11$$ $$Kgs$$ and their total value is maximized. Moreover, no item may be split. The total value of items picked by an optimal algorithm is denoted by $$V$$_opt. A greedy algorithm sorts the items by their value-to-weight ratios in descending order and packs them greedily, starting from the first item in the ordered list. The total value of items picked by the greedy algorithm is denoted by $$V$$_greedy.

The value of $$V$$_opt $$−$$ $$V$$_greedy is ____________.

Consider the following undirected graph $$G: $$

Choose a value for $$x$$ that will maximize the number of minimum weight spanning trees $$(MWSTs)$$ of $$G.$$ The number of $$MWSTs$$ of $$G$$ for this value of $$x$$ is ______.