Discrete Mathematics Answers

Solve the following. (35 pts)

1.    How many 3-digit number can be formed from digits 1 – 5 if:

a.   If repetition is allowed? (3 pts)

b.   If repetition is not allowed? (3 pts)

2.    How many strings of length 4 can be formed using the letters

COMPUTER if repetitions are not allowed? (3 pts)

3.    Out of 17 regions in the country, ten will be chosen to be included in a survey. How many ways of selecting 10 out of 17 regions? (3 pts)

4.    Expand the (2𝑥 − 𝑦)³ using binomial coefficient. (14 pts)

5.    Find for the coefficient of the following: (3 pts each)

a.   x⁴y³ after the expansion of (x – 2y )⁷

b.   x⁶y⁶ after the expansion of (2x + y )¹²

6.    Find the 9^th term of the expression (2x + y )¹² (3 pts)

BINOMIAL COEFFICIENTS (20 pts)

1.    Expand the (2𝑚 − 2𝑏)³ using binomial coefficient. (14 pts)

2.    Find for the coefficient of a⁵b⁵_; (a - 4b )¹⁰ (3 pts)

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

PIGEONHOLE PRINCIPLE

Show that in a group of 27 English words, there must be at least two that begin with the same letter. (5 pts) 

COMBINATIONS (24 pts)

1.    Patrick has assignments in 5 subjects. He can only do two assignments. In how many ways can he do two assignments? (3 pts)

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 pts)

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? (3 pts)

b.   They are all red? (3 pts)

c.    Two are blue and one is white? (3 pts)

d.   Exactly 2 are blue? (3 pts)

e.    None is white? (3 pts)

f.      At least two are white? (3 pts) 

PERMUTATIONS (21 pts)

1.    There are 6 people to be arranged in a line for a concert. How many arrangements are possible? (3 pts)

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

a.   Repetitions are not allowed? (3 pts)

b.   Repetitions are allowed? (3 pts)

c.    Starts with letter L and repetition is not allowed? (3 pts)

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? (3 pts)

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? (3 pts)

5.    Find the number of permutations of the word INFORMATION. (3 pts)

COUNTING METHODS (15 pts)

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

a.   If repetition is allowed? (3 pts)

b.   If repetition is not allowed? (3 pts)

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

(3 pts)

d.   If one (1) is not to be used as the 1^st digit and repetition is allowed? (3 pts)

Apply your knowledge gaining from this course in describing how you convert a function into a relation and a relation into a function?

Let S = {a₁,a₂,a₃,...a_n}be a set of test scores. Prove using the the indirect method of proof that if the average of this set of test scores is greater than 90, then at least one of the scores is greater than 90.

For each of these relations on the set {1, 2, 3, 4}, decide

whether it is reflexive, whether it is symmetric, whether

it is antisymmetric, and whether it is transitive.

 {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)}

Consider the following relation R on A where A = {1, 2, 3, 4, 5}

aRb ⇔

a

b

< min(a, b)

For example, 2R4 since

2

4

=

1

2

and min(2, 4) = 2 and 1

2

< 2.

(a) Draw the digraph of R (4)

(b) Give a path of length 2 from 3, if any (2)

(c) Give the domain and range of R. (4)

(d) Determine R(2)

a_n = a _0.5n + n, a₁ = 0, where n is a power of 2, is a linear recurrence relation. ( true / falsa ).