In software engineering, binary code is the most basic form of computer code, made up of only 0s and 1s, which computers use to interpret and execute instructions. Every piece of data, from text to images, is ultimately represented using binary code, where each 0 or 1 has a specific value depending on its position. Understanding how binary sequences function is essential for developing efficient algorithms and ensuring the reliability of digital systems, particularly in security applications.
Imagine you’re a cybersecurity expert designing a high-tech security system for a top-secret intelligence agency. The agency uses binary-based locks and authentication mechanisms to protect sensitive information. To ensure maximum security, you must analyze binary patterns and probabilities, creating a system that is both functional and resistant to hacking attempts.
4.1 You’re designing a binary code lock for a keypad. The lock requires a 6-digit code, where the first digit must be 1, and all remaining digits can be either 0 or 1. How many unique combinations of digits are possible for the code lock?
4.2 If someone attempts to open a 4-digit code lock by guessing randomly either a 0 or 1 for each position, what is the probability that they will guess the entire 4-digit code correctly on their first attempt? Express your answer as a common fraction.
4.3 You introduce an 8-digit authentication code where exactly three of the digits must be 1s, and the remaining five must be 0s. How many different 8-digit codes meet this requirement?
4.4 You have created the following four rows and four columns of binary code, where x, y and z are unknown digits that may be either 0 or 1. If the sum of all digits in Columns B and D equal 5, and two rows each have digit sums of 4, what is the value of z?
This problem set was submitted by a MATHCOUNTS volunteer, Brody Fogleman. Thank you, Brody!