Permutations and Combinations

The value of $b_6$ is

Which of the following is correct?

The number of A in $T_p$ such that A is either symmetric or skew-symmetric or both, and det (A) divisible by p is

The number of A in $T_p$ such that the trace of A is not divisible by p but det (A) is divisible by p is

[Note: The trace of a matrix is the sum of its diagonal entries.]

The number of A in $T_p$ such that det (A) is not divisible by p is

Let $S=\{1,2,3,4\}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is

The number of matrices in $\mathcal A$ is

Consider all possible permutations of the letters of the word ENDEANOEL.

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is