Questions in Section Discrete Mathematics

Consider a sequence of 10 A's and 8 B's placed in a row.

By a run we mean one or more letters of the same type placed side by side.

Here is an arrangement of 10 A's and 8 B's which contains 4 runs of A and 4 runs of B: AAABBABBBAABAAAABB.

Find the number of ways to travel from (0,0) to (5,7) if any unit steps in the positive x direction and the positive y direction are allowed, except that it is not possible to travel from (3,2) to (4,2).

Consider sequences of length 8 formed from 2 a's, 2 b's, 2 c's and 2 d's.

1. How many of these have no consecutive a's?
2. How many of these have 2 consecutive a's and 2 consecutive b's?

If the letters of the word ALGEBRA are randomly arranged in a sequence (of 7 letters), what is the probability that the two A's are together.

In how many ways can n married couples line up for a photograph if every wife stands next to her own husband?

What is the probability that a random 3n-digit ternary sequence has an equal number of 0's, 1's and 2's?

In how many ways can four blocks of four consecutive seats be chosen from 25 consecutive seats?

There are 12 chairs in a row on which we place five final exams, with at most one exam per chair. In how many ways this can be done, if no two adjacent chairs can have an exam?

Let X={1,2,3,4,5,6} and Y={2,4,6,8,10}.

In each of the following cases rho defines a binary operation from X to Y. For each one, list the elements of rho. (Recall that rho will be some subset of XxxY.)

A tree has precisely k vertices, where k>8. Exactly 5 of its vertices have degree 1, and exactly three of its vertices have degree 3. Find the degrees of all the remaining k-8 vertices (and explain your working).