Let a and b be positive real numbers. Suppose $$PQ = a\widehat i + b\widehat j$$ and $$PS = a\widehat i - b\widehat j$$ are adjacent sides of a parallelogram PQRS. Let u and v be the projection vectors of $$w = \widehat i + \widehat j$$ along PQ and PS, respectively. If |u| + |v| = |w| and if the area of the parallelogram PQRS is 8, then which of the following statements is/are TRUE?

For non-negative integers s and r, let

$$\left( {\matrix{ s \cr r \cr } } \right) = \left\{ {\matrix{ {{{s!} \over {r!(s - r)!}}} & {if\,r \le \,s,} \cr 0 & {if\,r\, > \,s} \cr } } \right.$$

For positive integers m and n, let

$$g(m,\,n) = \sum\limits_{p = 0}^{m + n} {{{f(m,n,p)} \over {\left( {\matrix{ {n + p} \cr p \cr } } \right)}}} $$

where for any non-negative integer p,

$$f(m,n,p) = \sum\limits_{i = 0}^p {\left( {\matrix{ m \cr i \cr } } \right)\left( {\matrix{ {n + i} \cr p \cr } } \right)\left( {\matrix{ {p + n} \cr {p - i} \cr } } \right)} $$

Then which of the following statements is/are TRUE?

An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021 is ...........

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is ..........