本次美国代写内容为离散数学的homework作业

乘法与加法原理与包含与排除原理
对于 1-6 题,你应该只使用乘法、加法和/或包含和排除的原则来得出你的答案(对应于文本中的第 6.1 节)。

将每个答案作为表达式和表达式的整数计算给出。

1. Tina 想为她的手机添加一个 PIN 码。她可以添加一个由 0-9 位数字组成的 PIN 码,长度从至少 4 位到最多 17 位。

(a) Tina 可以选择多少个长度为 5 的 PIN?

(b) Tina 认为更安全的密码不会重复任何数字。多少个 PIN
长度 5 Tina 可以创建唯一的数字吗?

(c) Tina 最喜欢的数字是 3 和 5。她有多少种方法可以选择长度为 5 的 PIN
以 3 或 5 开头,以相反的数字结尾(即,如果以 3 开头,则以 5 结尾,反之亦然)?

(d) Tina 不喜欢数字 7。她可以有多少种方式选择长度为 5 且不包含 7 的 PIN?

(e) Tina 最多只能记住 6 位数字,因此她只能选择长度为 4、5 或 6 的 PIN。她可以选择多少种方式来选择她能记住的 PIN?

2. 某大学生成6个字符的学生证如下:

• 第一个字符对应学生姓名的第一个字母
• 第二个字符对应学生姓氏的第一个字母
• 其余 4 个字符是十进制数 (0-9)。

(a) 大学可以生成多少个唯一 ID?

(b) 大学可以为一个特定的人生成多少个唯一 ID?

(c) 大学希望确保他们可以创建足够的唯一 ID。他们修改
他们的 ID 系统略有不同,因此现在您可以按如下方式放置两个字母: (1) 两个字母
在开头,(2)第一个字母在开头,第二个在结尾,或
(3) 两个字母都在末尾。该字母始终与第一个首字母 rst 一起放置,并且
最后一个首字母秒(例如,对于首字母为 A.B. 的学生,他们的 ID 可以为 AB、A B、
或AB)。他们可以使用这个新系统创建多少个唯一 ID?

3. 考虑集合 D 和 C,其中 jDj = 3 和 jCj = 8。

(a) 我们可以创建多少个唯一的有序对 (d; c),其中 d 2 D 和 c 2 C 是多少?
(b) 我们可以从 D 到 C 创建多少个独特的函数?
(c) 从 D 到 C,我们可以创建多少个唯一的 1-1 函数?

4. 琳达正在厨房里整理她的调料架。她有 12 个香料瓶和 2 个货架。琳达想在第一个货架上排列 7 种香料。

(a) 她可以用多少种不同的方式将 12 种香料中的 7 种放在第一个架子上?

(b) 琳达经常使用肉桂和豆蔻,所以她希望它​​们可以放在第一个架子上。如何
如果豆蔻和肉桂都是她的第一个架子,她可以通过多种方式在她的第一个架子上排列7种香料
包括?

(c) 琳达决定将肉桂或豆蔻放在第一个架子上,但不能同时放两者。她有多少种方法可以将她的第一个架子放在两种香料(肉桂和豆蔻)中的一种,但不能同时装两种。

Multiplication and Addition Principles & Principle of Inclusion Exclusion
For problems 1-6, you should only use the multiplication principle, the addition principle and/or the principle inclusion exclusion to arrive at your answers (corresponding to section 6.1 in the text).

Give each answer as both an expression and an integer evaluation of the expression.

1. Tina wants to add a PIN on her cellphone. She can add a PIN consisting of digits 0-9, of any length from at least 4 digits to at most 17.

(a) How many PINs of length 5 can Tina choose?

(b) Tina believes that a more secure password does not repeat any digits. How many PINs of
length 5 can Tina create that do not repeat digits?

(c) Tina’s favorite numbers are 3 and 5. How many ways can she choose a PIN of length 5 that
starts with either a 3 or a 5, and ends with the opposite digit (i.e. if it starts with 3, it ends with 5 and vice versa)?

(d) Tina does not like the number 7. How many ways can she choose a PIN of length 5 that does not contain a 7?

(e) Tina can only remember numbers of at most 6 digits, thus she can only choose a PIN of length 4,5, or 6. How many ways can she choose a pin that she can remember?

2. A university generates 6 character student ID’s in the following manner:

• The rst character corresponds to the rst letter of the students rst name
• The second character corresponds to the rst letter of the students last name
• The remaining 4 characters are decimal digits (0-9).

(a) How many unique ID’s can the university generate?

(b) How many unique ID’s can the university generate for a speci c person?

(c) The university wants to ensure that they are able to create enough unique ID’s. They modify
their ID system slightly so that the two letters can now be placed as follows: (1) both letters
are at the beginning, (2) the rst letter is at the beginning and the second is at the end, or
(3) both letters are at the end. The letters are always placed with the rst initial rst, and
last initial second (e.g. for a student with initials A.B., they could have ID’s AB , A B,
or AB). How many unique ID’s can they create with this new system?

3. Consider the sets D and C, where jDj = 3 and jCj = 8.

(a) How many unique ordered pairs (d; c) can we create where d 2 D and c 2 C?
(b) How many unique functions can we create from D to C?
(c) How many unique 1-1 functions can we create from D to C?

4. Linda is organizing her spice shelves in her kitchen. She has 12 spice bottles and 2 shelves to put them on. Linda wants to put 7 spices in a line on the rst shelf.

(a) In how many di erent ways can she arrange 7 of her 12 spices on the rst shelf?

(b) Linda uses cinnamon and cardamom a lot, so she wants them to be on the rst shelf. How
many ways can she arrange 7 spices on her rst shelf, if cardamom and cinnamon are both
included?

(c) Linda decides instead that she wants either cinnamon or cardamom on the rst shelf, but not both. How many ways can she arrange her rst shelf so that it contains one of the two spices (cinnamon and cardamom), but no both.

 


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