Let X be a square matrix. Consider the following two statements on X.

I. X is invertible.

II. Determinant of X is non-zero.

Which one of the following is TRUE?

Let G be an undirected complete graph on n vertices, where n > 2. Then, the number of different Hamiltonian cycles in G is equal to

Let U = {1, 2 ,..., n}. Let A = {(x, X) | x ∈ X, X ⊆ U}. Consider the following two statements on |A|.

I. |A| = n2^n–1

II. |A| = $$\sum\limits_{k = 1}^n {k\left( {\matrix{ n \cr k \cr } } \right)} $$

Which of the above statements is/are TRUE?

Suppose Y is distributed uniformly in the open interval (1,6). The probability that the polynomial 3x² + 6xY + 3Y + 6 has only real roots is (rounded off to 1 decimal place) _____.