$${P_1} = I = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right],\,{P_2} = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & 1 & 0 \cr } } \right],\,{P_3} = \left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right],\,{P_4} = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 \cr } } \right],\,{P_5} = \left[ {\matrix{ 0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 1 & 0 \cr } } \right],\,{P_6} = \left[ {\matrix{ 0 & 0 & 1 \cr 0 & 1 & 0 \cr 1 & 0 & 0 \cr } } \right]$$ and $$X = \sum\limits_{k = 1}^6 {{P_k}} \left[ {\matrix{ 2 & 1 & 3 \cr 1 & 0 & 2 \cr 3 & 2 & 1 \cr } } \right]P_k^T$$

where $$P_k^T$$ denotes the transpose of the matrix P_k. Then which of the following option is/are correct?

Let $$\overrightarrow a = 2\widehat i + \widehat j - \widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j + \widehat k$$ be two vectors. Consider a vector c = $$\alpha $$$$\overrightarrow a$$ + $$\beta $$$$\overrightarrow b$$, $$\alpha $$, $$\beta $$ $$ \in $$ R. If the projection of $$\overrightarrow c$$ on the vector ($$\overrightarrow a$$ + $$\overrightarrow b$$) is $$3\sqrt 2 $$, then the
minimum value of ($$\overrightarrow c$$ $$-$$($$\overrightarrow a$$ $$ \times $$ $$\overrightarrow b$$)).$$\overrightarrow c$$ equals ................

Let |X| denote the number of elements in a set X. Let S = {1, 2, 3, 4, 5, 6} be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number of ordered pairs (A, B) such that 1 $$ \le $$ |B| < |A|, equals .............

Suppose

det$$\left| {\matrix{ {\sum\limits_{k = 0}^n k } & {\sum\limits_{k = 0}^n {{}^n{C_k}{k^2}} } \cr {\sum\limits_{k = 0}^n {{}^n{C_k}.k} } & {\sum\limits_{k = 0}^n {{}^n{C_k}{3^k}} } \cr } } \right| = 0$$

holds for some positive integer n. Then $$\sum\limits_{k = 0}^n {{{{}^n{C_k}} \over {k + 1}}} $$ equals ..............