For n > 2, let a {0, 1}ⁿ be a non-zero vector. Suppose that x is chosen uniformly at random from {0, 1}ⁿ.
Then, the probability that $$\sum\limits_{i = 1}^n {{a_i}{x_i}} $$ is an odd number is _______.

Graph G is obtained by adding vertex s to K_3,4 and making s adjacent to every vertex of K_3,4. The minimum number of colours required to edge-colour G is _____.

Consider the functions

I. $${e^{ - x}}$$

II. $${x^2} - \sin x$$

III. $$\sqrt {{x^3} + 1} $$

Which of the above functions is/are increasing everywhere in [0,1]?

Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is ______.