Discrete Mathematics Answers

COMBINATIONS (mod4)

1. Patrick has assignments in 5 subjects. He can only do two assignments. In how many ways can he do two assignments?

2. In how many ways can a group of 5 men and 3 women be made out of a total of 10 men and 6 women?

3. A box contains 6 red, 5 blue and 3 white balls. In how many ways can we select 3 balls such that

a. They are of different colors?

b. They are all red?

c. Two are blue and one is white?

d. Exactly 2 are blue?

e. None is white?

f. At least two are white?

BINOMIAL COEFFICIENTS

1. Expand the (2𝑚 − 2𝑏)3 using binomial coefficient.

2. Find for the coefficient of a5b5 ; (a - 4b )10

3. Find the 5th term after expanding the expression (3x – 4y)15

PIGEONHOLE PRINCIPLE

Show that in a group of 27 English words, there must be at least two that begin with the same letter.

PERMUTATIONS (mod4)

1. There are 6 people to be arranged in a line for a concert. How many arrangements are possible?

2. How many strings of length 5 can be formed using the letters QUALITY if

a. Repetitions are not allowed?

b. Repetitions are allowed?

c. Starts with letter L and repetition is not allowed?

3. A group of 25 people are going to run a race. The top three runners earn gold, silver, and bronze medals. How many arrangements are possible?

4. In how many different ways can the letters of the word "CHANGE" be arranged in such a way that the vowels always come together?

5. Find the number of permutations of the word INFORMATION.

COUNTING METHODS mod4:

1. How many 5-digit number can be formed from digits 0 – 6 if:

a. If repetition is allowed?

b. If repetition is not allowed?

c. If one (1) is not to be used as the 1st digit and repetition is not allowed?

d. If one (1) is not to be used as the 1st digit and repetition is allowed?

2. How many possible passwords are there for the following conditions: 3 digits followed by 2 letters followed by 4 digits?

Show that (p → q) ∧ (q → r) → (p → r) is a tautology

Show that (p → q) ∧ (p → r) and p → (q ∧ r) are logically equivalent

Show that ¬(p ⊕ q) and p ↔ q are logically equivalent

Construct a truth table for each of these compound propositions.

a) p ∧ ¬p

b) p ∨ ¬p

c) (p ∨ ¬q) → q

d) (p ∨ q) → (p ∧ q)

e) (p → q) ↔ (¬q → ¬p)

f) (p → q) → (q → p)

Find all primes less than a specified positive integer n. (let’s say n =100).

(¬p→r) ∧ (q ↔p)

Use set builder notation to give a description of each of these sets.

a) {3, 6, 9, 12}

b) {−4,−3,−2,−1, 0, 1, 2, 3, −4}