1) The game Mastermind consists of one player creating a hidden code and the other player attempting to guess the code. A code consists of a sequence of four round pegs and each peg can be one of six different colors (red, orange, yellow, green, blue, purple). For example, the player making the code might choose "Blue, Purple, Green, Blue" (colors can repeat). When guessing, the colors and the sequence must match, meaning the guess "Purple, Blue, Blue, Green" would be incorrect. a) How many different codes can be formed? b) How many codes contain four different colors? c) What is the probability that a random code contains no red pegs? d) What is the probability that a random code contains exactly three pegs of the same color?

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Logic and Proofs

Discrete mathematics deals with the separated values of data such as integers. It is widely used in the practical field of computer science and mathematics. By knowing discrete mathematics one can improve their reasoning and problem-solving capabilities. Discrete mathematics became popular after the boom of computers which operate in discrete steps and store data in discrete bits. It also useful in studying computer programming, algorithms, and cryptography.