Discrete Mathematics Answers

a. How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of four aces (A) are chosen?

1. Discuss ways in which the current telephone numbering plan can be extended to accommodate the rapid demand for more telephone numbers. (See if you can find some of the proposals coming from the telecommunications industry.) For each new numbering plan you discuss, show how to find the number of different telephone numbers it supports.
2. Describe at least one way to generate all the partitions of a positive integer n. (You can get idea from Exercise 49 in Section 5.3.)
3. Proof that an undirected graph has an even number of vertices of odd degree.
4. In Exercises i) and ii) determine whether the given graph has a Hamilton circuit. If it does, find such a circuit. If it does not, give an argument to show why no such circuit exists.

Suppose a recurrence relation

an=2an−1−an−2

where a1=7 and a2=10

can be represented in explicit formula, either as:

Formula 1:

an=pxn+qnxn

              or  

Formula 2:

an=pxn+qyn

where 

x

and

y

are roots of the characteristic equation.

Determine p and q

 Answer:

p :

q :

Suppose that G is a connected multigraph with 2k vertices of odd degree. Show that there exist k subgraphs that have G as their union, where each of these subgraphs has a Euler path and where no two of these subgraphs have an edge in common.

1. Proof that an undirected graph has an even number of vertices of odd degree.

Describe at least one way to generate all the partitions of a positive integer n.

1. Discuss ways in which the current telephone numbering plan can be extended to accommodate the rapid demand for more telephone numbers. (See if you can find some of the proposals coming from the telecommunications industry.) For each new numbering plan you discuss, show how to find the number of different telephone numbers it supports.

Prove the statement by contraposition, if a product of two positive real numbers is greater

than 100, then at least one of the numbers is greater than 10.

For all integers n and m, if n − m is even then n^3 − m^3

is even.

An investor is considering a $25,000 investment in a start-up company. She estimates that she has

probability 0.05 of a $15,000 loss, probability 0.15 of a $20,000 loss, probability 0.35 of a $35,000 profit,

and probability 0.45 of breaking even (a profit of $0). What is the expected value of the profit?