Discrete Mathematics Answers

(a) How many code words over a, b, c, d of length 20 contain exactly 10 a’s?

(b) How many contain exactly 10 a’s and 5b’s.

There are 30 pupils in a class. (a) You make a row of 7 pupils. In how many ways can this be done? (b) You divide the class into two groups of 15 each. In how many ways can this be done? (c) You give each of the 30 pupils one type of cooldrink from 5 different types of cooldrink. In how many ways can this be done? (d) Ignoring who gets which cooldrink, how many different cooldrink combinations are possible if you choose 30 cooldrinks from 5 types?

Write notes on how to find cartesian products of sets and in you concluding page, site examples on the applications of set theory to solving real world business problems

(A∪B)

c

(A∪B)c

Determine whether the function f is a bijection from R to R ? Find fog and gof where f(x)=2x²+3 and g(x)=x=1

A. COUNTING METHODS (5 pts each)

1. How many strings of length 4 can be formed using the letters ABCDE if it starts with letters AC and repetition is not

allowed?

2. There are 10 multiple choice questions in an examination. Each of the questions have four choices. In how many ways

can an examinee give possible answers?

B. BINOMIAL COEFFICIENTS

Expand (2𝑥 + 4𝑎) 4 using the binomial theorem. (10 pts)

C. PIGEONHOLE PRINCIPLE (5 pts) . Explain briefly.

Do you agree that there are 3 persons who have the same first and last name? Why and why not? 

The set of all natural number whose square is more than 21. 

A = Q, (a*b) = a + b

What is the Cartesian product A × B × C, where A is the set of all airlines and B and C are both set of allcities in USA? Give an example of how this Cartesian product can be used

Identify the error or errors in this argument that supposedly shows that if ∃xP

(x) ∧ ∃xQ(x) is true then ∃x(P (x) ∧ Q(x)) is true.

a) ∃xP (x) ∨ ∃xQ(x) Premise

b) ∃xP (x) Simplification from (1)

c) P (c) Existential instantiation from (2)

d) ∃xQ(x) Simplification from (1)

e) Q(c) Existential instantiation from (4)

f) P (c) ∧ Q(c) Conjunction from (3) and (5)

g) ∃x(P (x) ∧ Q(x)) Existential generalization