Counting passwords made up of letters, digits, and special characters. Consider the following definitions for sets of characters: Digits = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Letters = { a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z) Special characters = { *, &, $, # } Compute the number of passwords that satisfy the given constraints. (a) Strings of length 6. Characters can be special characters, digits, or letters. (b) Strings of length 7, 8, or 9. Characters can be special characters, digits, or letters. (c) Strings of length 7, 8, or 9. Characters can be special characters, digits, or letters. The first character cannot be a letter. Exercise 6.1.2: Selecting lunch specials for the week. (a) A Chinese restaurant offers 10 different lunch specials. Each weekday for one week, Fiona goes to the restaurant and selects a lunch special. How many different ways are there for her to select her lunches for the week? Note that which lunch she orders on which day matters, so the following two selections are considered different. One possible selection: Mon: Kung pao chicken Tues: Beef with broccoli Wed: Kung pao chicken Thurs: Moo shu pork Fri: Beef with broccoli A different selection: Mon: Beef with broccoli Tues: Kung pao chicken Wed: Kung pao chicken Thurs: Moo shu pork Fri: Beef with broccoli bry strings (b) Now suppose that in addition to selecting her main course, she also selects between water or tea for her drink. How many ways are there for her to select her lunches?<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n Each character in the password can be one of the following:<\/p>\n\n\n\n This gives a total of: For each length, every character has 40 choices. We calculate: [ [ [ Summing them: Thus, there are 268.86 trillion<\/strong> possible passwords of length 7, 8, or 9.<\/p>\n\n\n\n The first character cannot<\/strong> be a letter, so we only have:<\/p>\n\n\n\n The remaining characters (positions 2 onward) still have 40 choices.<\/p>\n\n\n\n [ [ [ [ Summing them: Thus, there are 94.10 trillion<\/strong> possible passwords when the first character is restricted.<\/p>\n\n\n\n Fiona has 10 lunch choices per day<\/strong> for 5 days. Since order matters, we use: So, there are 100,000<\/strong> different ways for Fiona to pick her lunches.<\/p>\n\n\n\n Each day, she picks one of 10 meals<\/strong> and one of 2 drinks<\/strong> (water or tea). This means she has: Thus, there are 3.2 million<\/strong> different ways for Fiona to select her lunch with a drink option.<\/p>\n\n\n\n Counting passwords made up of letters, digits, and special characters. Consider the following definitions for sets of characters: Digits = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Letters = { a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-190754","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190754","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=190754"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190754\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=190754"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=190754"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=190754"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Explanation:<\/h3>\n\n\n\n
(a) Passwords of Length 6:<\/h4>\n\n\n\n
\n
[
10 + 26 + 4 = 40 \\text{ choices per character}
]
For a 6-character password:
[
40^6 = 4,096,000,000
]
Thus, there are 4.096 billion<\/strong> possible passwords of length 6.<\/p>\n\n\n\n(b) Passwords of Length 7, 8, or 9:<\/h4>\n\n\n\n
[
40^7 + 40^8 + 40^9
]<\/p>\n\n\n\n
40^7 = 16,384,000,000
]<\/p>\n\n\n\n
40^8 = 655,360,000,000
]<\/p>\n\n\n\n
40^9 = 26,214,400,000,000
]<\/p>\n\n\n\n
[
16,384,000,000 + 655,360,000,000 + 26,214,400,000,000 = 268,861,440,000,000
]<\/p>\n\n\n\n(c) Passwords of Length 7, 8, or 9 (Restricted First Character):<\/h4>\n\n\n\n
\n
Total = 14 choices<\/strong> for the first character.<\/li>\n<\/ul>\n\n\n\n
14 \\times 40^6 + 14 \\times 40^7 + 14 \\times 40^8
]<\/p>\n\n\n\n
14 \\times 4,096,000,000 = 57,344,000,000
]<\/p>\n\n\n\n
14 \\times 163,840,000,000 = 2,293,760,000,000
]<\/p>\n\n\n\n
14 \\times 6,553,600,000,000 = 91,750,400,000,000
]<\/p>\n\n\n\n
[
57,344,000,000 + 2,293,760,000,000 + 91,750,400,000,000 = 94,101,504,000,000
]<\/p>\n\n\n\n
\n\n\n\nLunch Specials:<\/h3>\n\n\n\n
(a) Without Drink:<\/h4>\n\n\n\n
[
10^5 = 100,000
]<\/p>\n\n\n\n(b) With Drink:<\/h4>\n\n\n\n
[
10 \\times 2 = 20 \\text{ choices per day}
]
For 5 days:
[
(10 \\times 2)^5 = 20^5 = 3,200,000
]<\/p>\n\n\n\n
\n\n\n\nFinal Answers:<\/h3>\n\n\n\n
\n