Please write out all the answers step-by-step and explain them in details:
Let's say you buy one ticket for the California Powerball lottery. This means that you can make a powerball selection. A powerball selection is composed of two parts: a subset of 5 integers from the set {1,2,3,...,69}; and 1 (powerball) integer from the set {1, 2, 3, ...,26) (the powerball integer could be the same as or different than any of the original 5 integers.) Examples of winning powerball selections this year so far: Date set of 5 Powerball Jan 23, 2021 {5,8,17, 27, 28} 14 Jan 20, 2021 {40, 53, 60, 68, 69} 22 Jan 16, 2021 {14, 20, 39, 65, 67} 2 Jan 13, 2021 {4, 19, 23, 25, 49} 14 Jan 9, 2021 {14, 26, 38, 45, 46} 13 Jan 6, 2021 {1,20, 22, 60, 66} Jan 2, 2021 {3, 4, 11, 41,67} 5 (a) How many different possible powerball selections are there? (please show your work.) (b) What is the minimum number of bits required (optimal encoding) to encode each powerball lottery ticket? (please show your work.) (c) There are 100 winning powerball selections selected each year (two times per week.) A winning ticket expires after a year. Suppose you are the new lottery commissioner and you are in charge of storing the last 100 winning powerball selections so that if somebody shows up with a winning ticket, you can validate it with the information from the last 100 winning powerball selections. 3 i. Using the current system, the powerball selections are stored using 8 bit ASCII by encod- ing each integer as two digits. For example, the winning powerball selection from Jan 2 is ({3,4, 11, 41,67} 5) and this powerball selection will be stored as: 030411416705. How many bits are required to store 100 powerball selections using the current system (using 8-bit ASCII)? ii. How many bits are required to store 100 powerball selections using the theoretical optimal encoding? (d) i. Develop an encoding process that uses the theoretical optimal number of bits. (Hint: convert the subset into fixed density binary string and use ranking.) ii. Based on your encoding algorithm, encode the powerball selection: ({1,2,3,4,5}, 1) Let's say you buy one ticket for the California Powerball lottery. This means that you can make a powerball selection. A powerball selection is composed of two parts: a subset of 5 integers from the set {1,2,3,...,69}; and 1 (powerball) integer from the set {1, 2, 3, ...,26) (the powerball integer could be the same as or different than any of the original 5 integers.) Examples of winning powerball selections this year so far: Date set of 5 Powerball Jan 23, 2021 {5,8,17, 27, 28} 14 Jan 20, 2021 {40, 53, 60, 68, 69} 22 Jan 16, 2021 {14, 20, 39, 65, 67} 2 Jan 13, 2021 {4, 19, 23, 25, 49} 14 Jan 9, 2021 {14, 26, 38, 45, 46} 13 Jan 6, 2021 {1,20, 22, 60, 66} Jan 2, 2021 {3, 4, 11, 41,67} 5 (a) How many different possible powerball selections are there? (please show your work.) (b) What is the minimum number of bits required (optimal encoding) to encode each powerball lottery ticket? (please show your work.) (c) There are 100 winning powerball selections selected each year (two times per week.) A winning ticket expires after a year. Suppose you are the new lottery commissioner and you are in charge of storing the last 100 winning powerball selections so that if somebody shows up with a winning ticket, you can validate it with the information from the last 100 winning powerball selections. 3 i. Using the current system, the powerball selections are stored using 8 bit ASCII by encod- ing each integer as two digits. For example, the winning powerball selection from Jan 2 is ({3,4, 11, 41,67} 5) and this powerball selection will be stored as: 030411416705. How many bits are required to store 100 powerball selections using the current system (using 8-bit ASCII)? ii. How many bits are required to store 100 powerball selections using the theoretical optimal encoding? (d) i. Develop an encoding process that uses the theoretical optimal number of bits. (Hint: convert the subset into fixed density binary string and use ranking.) ii. Based on your encoding algorithm, encode the powerball selection: ({1,2,3,4,5}, 1)
