6.1 WorksheetExercise 6.1.1: Counting passwords made up of letters, digits, and specialcharacters.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, I, 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.%3D%3D%3D(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 charactercannot 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 therestaurant and selects a lunch special. How many different ways are there for her to select her lunches forthe week? Note that which lunch she orders on which day matters, so the following two selections areconsidered different.A different selection:One possible selection:Mon: Beef with broccoliMon: Kung pao chickenTues: Kung pao chickenWed: Kung pao chickenThurs: Moo shu porkFri: Beef with broccoliTues: Beef with broccoliWed: Kung pao chickenThurs: Moo shu porkFri: Beef with broccoli(b) Now suppose that in addition to selecting her main course, she also selects between water or tea for herdrink. How many ways are there for her to select her lunches? 6.2 WorksheetExercise 6.2.1: Devising a 3-to-1 correspondence., 2,..., 30) to (1, 2, ..., 10) that is a 3-to-1 correspondence. (You mayfind that the division, ceiling or floor operations are useful.)Exercise 6.2.2: Using the bijection rule to count palindromes.If x is a string, then x is the reverse of the string. For example, if x = 1011, then x = 1101. A string is apalindrome if the string is the same backwards and forwards (i.e., if x = xR). Let B = (0, 1}. The set B is the setof all n-bit strings. Let Pn be the set of all strings in B" that are palindromes.(a) Show a bijection between P6 and B3.%3D%3De,olgmexs ol(b) What is |P6|?(c) Determine the cardinality of P7 by showing a bijection between P7 and B" for some n.Exercise 6.2.3: Using the bijection rule to count binary strings with even parity.Let B = {0, 1}. B" is the set of binary strings with n bits. Define the set En to be the set of binary strings with nbits that have an even number of 1's. Note that zero is an even number, so a string with zero 1's (i.e., a stringthat is all 0's) has an even number of 1's.(a) Show a bijection between Bº and E10. Explain why your function is a bijection.%3DDOUre'hoo 1-otslnoitonugoiovifieoqomee1ol anenogasmo0 1-of slnoitonul en(b) What is |E10|?SuewSLExercise 6.2.4: Using the bijection rule to count ternary strings whose digits sum toa multiple of 3.et T = {0, 1, 2}. A string x E T" is said to be balanced if the sum of the digits is an integer multiple of 3.a) Show a bijection between the set of strings in T6 that are balanced and T5. Explain why your function is aijection.How many strings in T6 are balanced?

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6.1 WorksheetExercise 6.1.1: Counting passwords made up of letters, digits, and specialcharacters.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, I, 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.%3D%3D%3D(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 charactercannot 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 therestaurant and selects a lunch special. How many different ways are there for her to select her lunches forthe week? Note that which lunch she orders on which day matters, so the following two selections areconsidered different.A different selection:One possible selection:Mon: Beef with broccoliMon: Kung pao chickenTues: Kung pao chickenWed: Kung pao chickenThurs: Moo shu porkFri: Beef with broccoliTues: Beef with broccoliWed: Kung pao chickenThurs: Moo shu porkFri: Beef with broccoli(b) Now suppose that in addition to selecting her main course, she also selects between water or tea for herdrink. How many ways are there for her to select her lunches? 6.2 WorksheetExercise 6.2.1: Devising a 3-to-1 correspondence., 2,..., 30) to (1, 2, ..., 10) that is a 3-to-1 correspondence. (You mayfind that the division, ceiling or floor operations are useful.)Exercise 6.2.2: Using the bijection rule to count palindromes.If x is a string, then x is the reverse of the string. For example, if x = 1011, then x = 1101. A string is apalindrome if the string is the same backwards and forwards (i.e., if x = xR). Let B = (0, 1}. The set B is the setof all n-bit strings. Let Pn be the set of all strings in B&#34; that are palindromes.(a) Show a bijection between P6 and B3.%3D%3De,olgmexs ol(b) What is |P6|?(c) Determine the cardinality of P7 by showing a bijection between P7 and B&#34; for some n.Exercise 6.2.3: Using the bijection rule to count binary strings with even parity.Let B = {0, 1}. B&#34; is the set of binary strings with n bits. Define the set En to be the set of binary strings with nbits that have an even number of 1's. Note that zero is an even number, so a string with zero 1's (i.e., a stringthat is all 0's) has an even number of 1's.(a) Show a bijection between Bº and E10. Explain why your function is a bijection.%3DDOUre'hoo 1-otslnoitonugoiovifieoqomee1ol anenogasmo0 1-of slnoitonul en(b) What is |E10|?SuewSLExercise 6.2.4: Using the bijection rule to count ternary strings whose digits sum toa multiple of 3.et T = {0, 1, 2}. A string x E T&#34; is said to be balanced if the sum of the digits is an integer multiple of 3.a) Show a bijection between the set of strings in T6 that are balanced and T5. Explain why your function is aijection.How many strings in T6 are balanced?

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