Let $${A_1},\,{A_2},\,{A_3}$$ and $${A_4}$$ be four matrices of dimensions $$10 \times 5,\,5 \times 20,\,20 \times 10,$$ and $$10 \times 5,$$ respectively. The minimum number of scalar multiplications required to find the product $${A_1}{A_2}{A_3}{A_4}$$ using the basic matrix multiplication method is _________.

Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than $$100$$ hours given that it is of Type $$1$$ is $$0.7,$$ and given that it is of Type $$2$$ is $$0.4.$$ The probability that an LED bulb chosen uniformly at random lasts more than $$100$$ hours is _________.

Consider the system, each consisting of m linear equations in $$n$$ variables.
$$I.$$ $$\,\,\,$$ If $$m < n,$$ then all such system have a solution
$$II.$$ $$\,\,\,$$ If $$m > n,$$ then none of these systems has a solution
$$III.$$ $$\,\,\,$$ If $$m = n,$$ then there exists a system which has a solution

Which one of the following is CORRECT?

Consider the following expressions:
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$(i)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ false
$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $$(ii)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$Q$$
$$\,\,\,\,\,\,\,\,\,\,\,$$ $$(iii)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ true
$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $$(iv)$$ $$\,\,\,\,\,\,\,\,\,\,\,$$ $$P∨Q$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$(v)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\neg QVP$$

The number of expressions given above that are logically implied by $$P \wedge \left( {P \Rightarrow Q} \right)$$) is _____________.