Question Solved1 Answer Question 2: A password is a string of 8 characters taken from the alphabet (that is, the set of 26 characters \( \{a, b, c, \ldots, x, y, z\}) \), the set of 10 digits \( \{0,1,3,4,5,6,7,8,9\} \) and the set of 5 special characters \( \{!, @, \#, *,+\} \). a) How many unique passwords are there? b) How many unique passwords are there where no character can be repeated and there are exactly 2 digits? c) How many unique passwords are there where you can repeat characters and there are at least 2 digits? d) How many unique passwords are there where you can repeat characters and have at least 2 digits and at least 1 special character?

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Transcribed Image Text: Question 2: A password is a string of 8 characters taken from the alphabet (that is, the set of 26 characters \( \{a, b, c, \ldots, x, y, z\}) \), the set of 10 digits \( \{0,1,3,4,5,6,7,8,9\} \) and the set of 5 special characters \( \{!, @, \#, *,+\} \). a) How many unique passwords are there? b) How many unique passwords are there where no character can be repeated and there are exactly 2 digits? c) How many unique passwords are there where you can repeat characters and there are at least 2 digits? d) How many unique passwords are there where you can repeat characters and have at least 2 digits and at least 1 special character?
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Transcribed Image Text: Question 2: A password is a string of 8 characters taken from the alphabet (that is, the set of 26 characters \( \{a, b, c, \ldots, x, y, z\}) \), the set of 10 digits \( \{0,1,3,4,5,6,7,8,9\} \) and the set of 5 special characters \( \{!, @, \#, *,+\} \). a) How many unique passwords are there? b) How many unique passwords are there where no character can be repeated and there are exactly 2 digits? c) How many unique passwords are there where you can repeat characters and there are at least 2 digits? d) How many unique passwords are there where you can repeat characters and have at least 2 digits and at least 1 special character?
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Step1/4.gkwtCW{margin:0;font-family:"Aspira Webfont","Helvetica","Arial",sans-serif;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-direction:column;-ms-flex-direction:column;flex-direction:column;gap:16px;}/*!sc*/data-styled.g379[id="sc-z3f5s1-0"]{content:"gkwtCW,"}/*!sc*/.iIwMoS{white-space:pre-wrap;}/*!sc*/data-styled.g381[id="sc-1aslxm9-0"]{content:"iIwMoS,"}/*!sc*/.fzJtOB{text-align:start;}/*!sc*/data-styled.g383[id="sc-1aslxm9-2"]{content:"fzJtOB,"}/*!sc*/.hOZehF{margin:0;font-family:"Aspira Webfont","Helvetica","Arial",sans-serif;}/*!sc*/data-styled.g410[id="sc-9wsboo-0"]{content:"hOZehF,"}/*!sc*/.lhIoTe{margin:0;font-size:1rem;}/*!sc*/data-styled.g412[id="sc-1swtczx-0"]{content:"lhIoTe,"}/*!sc*/.iHelzO{margin:0;font-family:"Aspira Webfont","Helvetica","Arial",sans-serif;line-height:normal;}/*!sc*/data-styled.g445[id="sc-1sugbjn-0"]{content:"iHelzO,"}/*!sc*/a) Answer: There are 26 characters in the alphabet, 10 digits, and 5 special characters, so there are 26+10+5=41 total characters to choose from. A password is 8 characters long, so there are 41^8 = 6,634,204,312,890,625 unique passwords.Explanation:Please refer to solution in this step.Step2/4.gkwtCW{margin:0;font-family:"Aspira Webfont","Helvetica","Arial",sans-serif;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-direction:column;-ms-flex-direction:column;flex-direction:column;gap:16px;}/*!sc*/data-styled.g379[id="sc-z3f5s1-0"]{content:"gkwtCW,"}/*!sc*/.iIwMoS{white-space:pre-wrap;}/*!sc*/data-styled.g381[id="sc-1aslxm9-0"]{content:"iIwMoS,"}/*!sc*/.fzJtOB{text-align:start;}/*!sc*/data-styled.g383[id="sc-1aslxm9-2"]{content:"fzJtOB,"}/*!sc*/.hOZehF{margin:0;font-family:"Aspira Webfont","Helvetica","Arial",sans-serif;}/*!sc*/data-styled.g410[id="sc-9wsboo-0"]{content:"hOZehF,"}/*!sc*/.lhIoTe{margin:0;font-size:1rem;}/*!sc*/data-styled.g412[id="sc-1swtczx-0"]{content:"lhIoTe,"}/*!sc*/.iHelzO{margin:0;font-family:"Aspira Webfont","Helvetica","Arial",sans-serif;line-height:normal;}/*!sc*/data-styled.g445[id="sc-1sugbjn-0"]{content:"iHelzO,"}/*!sc*/b) Answer: There are 10 digits to choose from, and we need to select exactly 2 of them. The remaining 6 characters can be any of the 41 total characters. So there are 10C2 = 45 ways to choose the 2 digits. For the remaining 6 characters, there are 414039383736 ways to choose them. So there are 4541^6 = 3,158,765,732,945,000 unique passwords where no character can be repeated and there are exactly 2 digits.Explanation:Please refer to solution in this step.Step3/4.gkwtCW{margin:0;font-family:"Aspira Webfont","Helvetica","Arial",sans-serif;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-direction:column;-ms-flex-direction:column;flex-direction:column;gap:16p ... See the full answer