Let $$U = \{ 1,2,....,n\} $$, where n is a large positive integer greater than 1000. Let k be a positive integer less than n. Let A, B be subsets of U with $$|A| = |B| = k$$ and $$A \cap B = \phi $$. We say that a permutation of U separates A from B if one of the following is true.

- All members of A appear in the permutation before any of the members of B.

- All members of B appear in the permutation before any of the members of A.

How many permutations of U separate A from B?

Let $$f:A \to B$$ be an onto (or surjective) function, where A and B are nonempty sets. Define an equivalence relation $$\sim$$ on the set A as

$${a_1} \sim {a_2}$$ if $$f({a_1}) = f({a_2})$$,

where $${a_1},{a_2} \in A$$. Let $$\varepsilon = \{ [x]:x \in A\} $$ be the set of all the equivalence classes under $$\sim$$. Define a new mapping $$F:\varepsilon \to B$$ as

$$F([x]) = f(x)$$, for all the equivalence classes $$[x]$$ in $$\varepsilon $$.

Which of the following statements is/are TRUE?

Let X be a set and 2$$^X$$ denote the powerset of X. Define a binary operation $$\Delta$$ on 2$$^X$$ as follows:

$$A\Delta B=(A-B)\cup(B-A)$$.

Let $$H=(2^X,\Delta)$$. Which of the following statements about H is/are correct?

Consider a random experiment where two fair coins are tossed. Let A be the event that denotes HEAD on both the throws, B be the event that denotes HEAD on the first throw, and C be the event that denotes HEAD on the second throw. Which of the following statements is/are TRUE?