1. Determine how many strings of n lowercase letters from the English alphabet contain

(a) the letter a.

(b) the letters a and b.

(c) the letters a and b in consecutive positions with a preceding b, with all letters of the string distinct.

(d) the letters a and b, where a is somewhere to the left of b in the string, with all letters distinct.

2. Five rooms of a house are to be painted in such a way that rooms with an interconnecting door have different colors. If there are n colors available, how many different color schemes are possible when the rooms in the house are arranged in the following way?

(a) Connected rooms form a linear order with one door interconnecting two adjacent rooms.

(b) Connected rooms form a linear order with one door interconnecting two adjacent rooms. The first and last rooms must be colored differently.

(c) Connected rooms form a circular order with one door interconnecting two adjacent rooms.