Discrete Mathematics for Computer Science

Exercises 435

24. A survey asks the respondent to order by importance 10 properties of a car. How many
    orderings are possible? How many orderings are there if the first and last property are
    given?


25. Find the number of paths from A to F in the following diagram with six letters. A path
    can only go through letters that are consecutive, either horizontally or vertically, and
    it goes only to the right or up at each step.

F

E F

D E F

C D E F

B C D E F

A B C D E F

Prove that a similar path with n letters has 2'- 1 paths from the lower left corner to any

letter in the rightmost position in a row.

Internet Addresses: IPv4 and IPv6. The Internet requires an address for each machine

that is connected to it. The address space of the addressing architecture of Internet Protocol

version 4 (IPv4) consists of a 32-bit field. Since not every combination of bits can be used

as an address, plans are underway to change the address space to a 128-bit field in IPv6.

The 32-bit IPv4 addresses are usually written in a form called dotted decimal. The 32-

bit address is broken up into four 8-bit bytes, and these bytes are then converted to their

equivalent decimal form and separated by dots. For example,

10000000 00000011 00000010 00000011

is written as 128.3.2.3, which is obviously more readable. The 128-bit IPv6 addresses
are divided into eight 16-bit pieces. Each 16-bit piece is converted to its equivalent
hexadecimal value (each sequence of 4 bits is converted to one hexadecimal digit).
The eight four-character hexadecimal strings are separated by colons. It is not prac-
tical to list 128 bits and show the conversion to the final IPv6 address form. As
an example of what you might end up with, however, we show one IPv6 address:
FEDC:BA98:7654:3210:FEDC:BA98:7654:3210.

26. How many IPv4 addresses are possible?


27. Write the following IPv4 address in dotted decimal format:

01001010 11001010 10001000 11011101

where XlX 2 X3X4X5x6X7x8 as a binary number is the decimal number

xl • 2^7 +X2 • 26 + x3 • 2^5 + x4 • 2^4 + x5 • 2^3 +x6. 22 + x7 • 2^1 + x8 • 2^0

28. How many IPv6 addresses are possible?


29. What would the string

01001000 11101010 01101001 01110111

look like as the first part of an IPv6 address?