1. Use Mathematical Induction to prove the following:

(i) n³ - n is divisible by 3, for all integers n ≥ 1;

(ii) 1 + 3 + 5 + ? + (2n - 1) = n² for all integers n ≥ 1.

2. How many positive integers between 100 and 699 inclusive

(i) have the same first and third digits?

(ii) are divisible by 4?

(iii) are divisible by 6? 

(iv) are divisible by both 4 and 6?

(v) are divisible by either 4 or 6?

(vi) are divisible by 6 but not by 4?

3. Given that A = {1, 2, 3, 4} and B = {5, 6, 7, 8, 9}, how many functions f: A → B are there
(i) in total?

(ii) which are one-to-one?

(iii) with f(1) = 5, f(2) = 6 and f(3) = 7?

(iv) with f(1) ≠ 5, and f(3) ≠ 7 ?

4. (i) At the local school there are 12 different time periods during which 370 different classes must be scheduled. How many rooms does the school need?

(ii) How many people must you have to ensure that at least 12 of them have the same birthday (assume 366 possible birthdays)?

5. A ship's crew is to be selected from 25 people. There are 5 captains, 7 leading hands, 5 midshipmen and 8 ordinary sailors available. If the crew is to consist of 1 captain, 2 leading hands, 3 midshipmen and 4 ordinary sailors, in how many different ways can the crew be formed?

6. Find the coefficient of x²⁷y² in the expansion of (-x - 3y)²⁹.

7. In how many different ways can a selection of 6 coloured balls be made from a barrel containing red, green, yellow and blue balls and the selection must contain at least 2 blue balls?

8. Using all the letters, find the number of different strings that can be formed from the word: ARKAROOLA

9. For each of the following relations R on the set A = {2, 4, 6, 8, 10}, determine (give reasons) whether R is:

(a) reflexive; (b) symmetric; (c) transitive (i) R = {(2,2), (2,4), (4,2), (4,6), (2,6), (6,4), (6,2)}

(ii) R = {(2,2), (4,4), (6,6), (8,8), (10,10), (2,4), (4,6), (2,6)}

(iii) R = {(2,10), (10,2), (4,8), (8,4), (6,6)}