PART A ( show the steps )

• 1. Calculate the number of integers divisible by 4 between 50 and 500, inclusive.

2. Hexadecimal digits are formed using either a numeric decimal digit or a letter from A to F. How many possible digits can be chosen? Did you use the sum rule or the product rule?

3. A combination lock requires three numbers from 1 to 25. How many combinations are possible? Did you use the sum rule or the product rule? (Note that although we use the term combination with locks, order matters.)

4. Use the permutation formula to calculate the number permutations of the set {a, b, c, d} taken two at a time. Also list these permutations.

5. Using the pigeonhole principle, show that any 11 digit decimal number must contain two instances of some decimal digit.

PART B

1. 6. Which of the following sentences are logical statements?
   1. a. If x even divides y, then x is a factor of y
   2. b. If John does well in discrete math, then he will be an excellent programmer
   3. c. 2 is the only even prime number
   4. d. He is the best student in the class
2. 7. Construct the truth tables for the following propositions:
   1. 1. (p p) q
   2. 2.(p q) (q p)
   3. 3. p (q r)
   4. 4. (p q) (p r)
3. 8. Refer to the propositions in problem 2. For each of them, indicate whether it is a tautology, a contradiction or neither.
4. 9. Use truth tables to determine whether the following is valid argument:

p q q p p q

• 10. Use truth tables to determine whether each of the following pairs of propositions are logically equivalent.

1. 1. a. (p q) q
   2. q (q p)
   3. b ( (p q)) q
   4. ( p q) q