Let $$X$$ and $$Y$$ denote the sets containing $$2$$ and $$20$$ distinct objects respectively and $$𝐹$$ denote the set of all possible functions defined from $$X$$ to $$Y$$. Let $$f$$ be randomly chosen from $$F.$$ The probability of $$f$$ being one-to-one is ________.

Which one of the following well formed formulae is a tautology?

The cardinally of the power set of $$\left\{ {0,1,2,\,\,....,\,\,10} \right.\left. \, \right\}$$ is _____________.

Perform the following operations on the matrix $$\left[ {\matrix{ 3 & 4 & {45} \cr 7 & 9 & {105} \cr {13} & 2 & {195} \cr } } \right]$$
(i) Add the third row to the second row
(ii) Subtract the third column from the first column.

The determinant of the resultant matrix is _________.