Discrete Mathematics Answers

Let n be 26. Construct a map of a continent having n different countries in such a way that four colours are needed to colour bordering countries in different colours. Using blue for the ocean (and possibly for some of the countries), and three other colours of your choice for the countries, apply colours to the map. [Use four colours whose names start with different letters, so that you can represent each colour with the first letter of its name.]

(b) By inserting vertices at strategic locations, convert your map into a connected planar graph in which faces separated by an edge are differently coloured.

(c) Construct the dual graph of your connected planar graph, and apply the corresponding vertex colouring to your dual graph

please i need answer in simple form with clear explanation to understand , i need answer as requirement

A. COUNTING METHODS

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.

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? 

In a market, 50 women sell only Apple, 25 sell Apple and Banana and 50 sell Banana. Each woman sells at least one the two fruits. How many women are there? 

The relation 'is the father of' is________.

a) reflexive b) irreflexive

c) transitive d) symmetric

Let p: priya is tall and q:priya is beautiful write the following statement in symbolic form.

i) Priya is tall and beautiful

ii) Priya is tall but beautiful

iii) It is false that priya is short or beautiful

. A special type of password consists of six(6) different letters of the alphabet, where each letter is used only once. How

many different possible passwords are there? (3 pts)

A. 244, 140, 625

B. 308, 915, 776

C. 165,765,600

D. 213,127,200 

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? 

If P(k) = k2

(k + 2)(k – 1) is true, then what is P (k + 1)? 

At a college of 600 students there are (n+5) enrolled in physics, (n+20) in Mathematics who play sports and (n-15) in chemistry and play sports. Create a Venn diagram to illustrate this information. Where n=8

At a college of 600 students there are (n+5) enrolled in physics, (n+20) in Mathematics who play sports and (n-15) in chemistry and play sports. Create a Venn diagram to illustrate this information. Where n is your arid number.