Colors in HTML (the language in which many web pages are...

Colors in HTML (the language in which many web pages are written) can be represented by six-digit hexadecimal codes: sequences of six integers ranging from 0 to 15 (represented as $0, \ldots, 9, \mathrm{~A}, \mathrm{~B}, \ldots, \mathrm{F})$ a. How many different colors can be represented? b. Some monitors can display only colors encoded with pairs of repeating digits (such as $44 \mathrm{DD} 88$ ). How many colors can these monitors display? c. Grayscale shades are represented by sequences $x y x y x y$ consisting of a repeated pair of digits. How many grayscale shades are possible? d. The pure colors are pure red: $x y 0000$; pure green: $00 x y 00$; and pure blue: $0000 x y .(x y=F F$ gives the brightest pure color, while $x y=00$ gives the darkest: black.) How many pure colors are possible?

Colors in HTML (the language in which many web pages are written) can be represented by six-digit hexadecimal codes: sequences of six integers ranging from 0 to 15 (represented as $0, \ldots, 9, \mathrm{~A}, \mathrm{~B}, \ldots, \mathrm{F})$
a. How many different colors can be represented?
b. Some monitors can display only colors encoded with pairs of repeating digits (such as $44 \mathrm{DD} 88$ ). How many colors can these monitors display?
c. Grayscale shades are represented by sequences $x y x y x y$ consisting of a repeated pair of digits. How many grayscale shades are possible?
d. The pure colors are pure red: $x y 0000$; pure green: $00 x y 00$; and pure blue: $0000 x y .(x y=F F$ gives the brightest pure color, while $x y=00$ gives the darkest: black.) How many pure colors are possible?

Key Concepts

Hexadecimal Numeral System

This is a base?16 counting system that uses sixteen distinct symbols: the numbers 0 through 9 and the letters A through F. It is widely used in computer science and web development, especially for representing color values in HTML and CSS, because its two?digit groups (ranging from 00 to FF) neatly map to the 256 intensity levels used by many digital systems.

Fundamental Counting Principle

A core concept in combinatorics, the fundamental counting principle states that if there are several stages of selection with a certain number of independent choices at each stage, the total number of outcomes is the product of the number of choices at each stage. This principle is key when calculating the number of possible color codes or variations in patterns where each digit or group of digits can be chosen independently.

Digital Color Representation

Digital color representation, particularly in web design, uses combinations of hexadecimal numbers to encode the intensities of the three primary colors (red, green, and blue). Each primary color is typically represented by two hexadecimal digits, allowing for 256 possible intensities per channel. Understanding this model helps anyone interpret how variations in the hexadecimal values lead to different colors, pure colors, or even grayscale shades.

Counting with Constraints

Often in combinatorial problems, restrictions are imposed that limit what choices can be made in the sequence. For example, when digits must appear in repeating pairs or specific patterns (such as repeated pairs in a six-digit code), the available choices are reduced relative to the unconstrained situation. This concept is crucial for determining the number of options available when certain parts of a color code are fixed or required to follow a pattern.

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