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  • 差异

    lesson content

    Naima and Michael are in charge of staffing the middle school Sock Hop.  They begin with the five 6th-graders who have volunteered to work:  Cindy, Justin, Willy, Donna, and Angelle.  They are trying to figure out the order of assignment.  Does order matter with regard to this situation?
    ::Naima和Michael负责为Sock Hop中学配备工作人员。他们首先从自愿工作的5名6年级学生开始:Cindy、Justin、Willy、Donna和Angell。他们正在试图弄清楚任务顺序。对于这种情况,命令是否重要?

    In this concept, you will learn about permutations.
    ::在这个概念中,你会了解变异。

    Permutations
    ::差异

    You can make all kinds of combinations. Let’s say that you are making a pizza with pepperoni, mushrooms and peppers. It doesn’t matter which order you put the toppings on the pizza. You will still have the same pizza.
    ::你可以做各种组合。让我们假设你正在用辣椒、蘑菇和辣椒做披萨。 命令你把图纸放在披萨上并不重要。 你仍然会有同样的披萨。

    Sometimes, order does make a difference. When you arrange different objects or events and order is important, this is called a .
    ::有时,秩序确实有区别。当您安排不同的物体或事件和秩序是重要的时,这被称为 。

    Consider the word CAT. Clearly, order is important when you spell a word. You can write all of the correct letters, but if you don’t put them in the correct order, you don’t spell CAT. For example, here are some orders of C, A, and T that don’t spell CAT;
    ::考虑一下CAT这个字。 显然,当您拼写一个字时,秩序是重要的。 你可以写出所有正确的字母,但如果您不按正确的顺序写,您就不会拼写CAT。 比如,这里有一些C、A和T的指令不会拼写CAT;

    Incorrect orders: A C T , A T C , C T A , T A C , T C A Correct orders: C A T

    ::不正确的命令:ACT、ATTC、CTA、TAC、TCA更正命令:CAT

    This is a situation where order makes a difference.  
    ::这是一种秩序不同的情况。

    Let's look at an example.
    ::让我们举个例子。

    Tomás wants to know how many 3-digit numbers he can write using the digits 7, 8, and 9 without repeating any of the digits. Does order matter for this problem?
    ::Tomás想知道他用数字7、8和9可以写多少个3位数的数字,而不重复任何数字。

    First, write out a single order:  789
    ::首先,写下单单单订单:789

    Next, rearrange the order:  798
    ::下一步,重新安排顺序:798

    Then decide if order matters:  This is a different number from the original number, so order does matter.  Each arrangement of digits is a different permutation.
    ::然后决定顺序是否重要:这是一个与原编号不同的数字,所以顺序很重要。数字的每种排列是不同的变换。

    Examples
    ::实例

    Example 1
    ::例1

    Earlier, you were given a problem about  Naima and Michael, who are assigning job duties for the middle school dance.
    ::早些时候,你被问及 奈玛和迈克尔的问题, 他们分配 工作职责在中学舞蹈。

    They are assigning entrance door duty, beginning with the 6th-graders: Cindy, Justin, Willy, Donna, and Angelle.  Does the order in which they are assigned to perform the duty matter?
    ::他们分配了入口值勤,从第6年级起:辛迪、贾斯汀、威利、唐娜和安格尔。 他们被指派履行该职责的顺序是否?

    First, write out a single order:
    ::首先,写出单单单单单:

    Cindy, Justin, Willy, Donna, Angelle
    ::辛迪、贾斯汀、威利、唐娜、安卓

    Next, rearrange the order:
    ::下一步,重新排列顺序:

    Willy, Donna, Cindy, Angelle, Justin
    ::威利,唐娜,辛蒂,安格尔,贾斯汀

    Then, decide if the order of the assignment matters:
    ::然后决定转让顺序是否重要:

    The answer is the order does not matter; the duties will be covered by a 6th-grader no matter what the order.  This is not a permutation. 
    ::答案是命令无关紧要;不管命令是什么,职责将由第6年级学生承担。这不是变换。

    Example 2
    ::例2

    Henry has a lock. He can program the combination of his lock using the numbers 1, 5 and 18. Henry can only choose one arrangement of the numbers. Is this a permutation? Why?
    ::Henry有一个锁。他可以用1号、5号和18号 来编程他的锁的组合。 Henry只能选择一个数字的组合。 这是变换吗? 为什么?

    First, write out a single order:  1, 5, 18
    ::首先,写下单单单单单单单: 1, 5, 18

    Next, rearrange the order:  5, 18, 1
    ::下一步,重新安排顺序:5、18、1

    Then, decide if order matters in this situation.
    ::那么,在这种情形下,决定顺序是否重要。

    The answer is order does matter in this situation since there is only one way to order the numbers that will operate the lock.  This is a permutation.
    ::答案是,在这种情况下,顺序确实很重要,因为只有一个方法可以命令使用锁的号码。这是一个变换。

    Example 3
    ::例3

    Write whether or not order is important and why:
    ::写下命令是否重要,为什么:

    The number of 3-letter words Brenda can write using the letters E , T , A   without repeating letters.
    ::布伦达用E,T,A字母可以写三个字母的字数,不用重复字母。

    First, write out a 3-letter word.
    ::首先,写个三字母单词

    T-E-A
    ::T-E-A (T-E-A)

    Next, rearrange the lineup:
    ::下一步, 重新排列阵列 :

    A-T-E
    ::A-T-E

    Then, decide if you change the outcome by rearranging the order.  If so, then order matters.
    ::然后决定您是否通过重新排列顺序来改变结果。 如果是的话,请按顺序排列。

    The answer is T-E-A is a different word from A-T-E, so order does matter.  This is an example of a permutation.
    ::答案是 T- E- A 是与 A- T- E 不同的单词, 因此命令很重要 。 这是变换的示例 。

    Example 4
    ::例4

    Write whether or not order is important and why:
    ::写下命令是否重要,为什么:

    Five different cars entered the race painted the following colors: red, orange, blue, white, purple? In how many different ways can the cars finish the race?
    ::进入赛场的五辆不同的汽车涂上以下颜色:红色、橙色、蓝色、白色、紫色。 汽车能用多少种不同的方式完成赛事?

    First, write out a single lineup:
    ::首先,写一个单行:

    purple, white, red, orange, blue
    ::紫、白、红、橙、蓝色

    Next, rearrange the lineup:
    ::下一步, 重新排列阵列 :

    red, blue, orange, purple, white
    ::红色、蓝色、橙色、紫色、白色

    Then, decide if you change the outcome? If so, then order matters.
    ::那么,决定你是否改变结果?如果是的话,请点顺序。

    The answer is the change in order results in a different winner, so order does matter.  This is a permutation.
    ::答案是顺序变化导致不同的赢家, 所以顺序很重要。 这是一个变换 。

    Example 5
    ::例5

    Write whether or not order is important and why:
    ::写下命令是否重要,为什么:

    At the breakfast buffet you can take any three of the following: eggs, pancakes, potatoes, cereal, waffles. How many different 3-item breakfasts can you get?
    ::在早餐自助餐上,你可以吃以下三种:鸡蛋、煎饼、土豆、麦片、华夫饼。你能吃多少不同的三点早餐?

    First, write out a single order: 
    ::首先,写出单单单单单:

    eggs-pancakes-potatoes
    ::鸡蛋煎饼蛋蛋

    Next, rearrange the order: 
    ::下一步,重新排列顺序:

    potatoes-eggs-pancakes
    ::马铃薯煎饼

    Then decide if order matters:  
    ::然后决定命令是否重要:

    The answer is in either order, you will get the same food.  So order does not matter.  This is not a permutation.
    ::答案是任意顺序的,你们将获得同样的食物。所以,命令无关紧要。这不是一种变换。

    Review
    ::回顾

    Decide whether or not order matters for each of the following scenarios. Briefly explain your reasoning.
    ::决定下命令是否对以下每一种情况都很重要。 请简要解释一下您的推理 。

    1. Doug is going to use the following 5 letters to create his new 3-letter computer password: B, F, G, L, and T. How many different passwords can he create if he doesn’t repeat any letters?
      ::Doug将使用以下5封信来创建他新的3字母电脑密码:B、F、G、L和T。如果他不重复任何信件,他能创造多少不同的密码?
    2. Violin players in the orchestra include Jerry, Kerry, Barry, Mary, Sherry, Harry, Terri, and Perry. How many different 3-person trios can you make?
      ::管弦乐团中的violin球员包括Jerry、Kerry、Barry、Mary、Sherry、Harry、Terri和Perry。你能做多少不同的3人三角曲?
    3. The 3 different numbers for Arun’s lock are 14, 35, and 20. How many different combinations must Arun try before he’ll be sure he can open his lock?
      ::阿伦的锁有三个不同的号码是14、35和20。 阿伦要尝试多少不同的组合才能确定他能打开锁?
    4. Mr. Chen has decided that he’s going to give Nikki, Mickey, and Hickey awards for the essay contest. What he doesn’t know is who will get 1st prize, 2nd prize, and 3rd prize. How many different ways can Mr. Chen give out the prizes?
      ::陈先生决定授予Nikki、Mickey和Hickey作文比赛奖项。 他不知道的是谁将获得第一奖、第二奖和第三奖。 陈先生能以多少种不同的方式颁发这些奖项?
    5. Five candidates are running for 2 student senate seats: Bo, Jo, Mo, Zo, and Ro. How many different pairs of senators can there be?
      ::五个候选人正在竞选两个参议院学生席位:博、乔、莫、佐和罗。 有多少不同的参议员?
    6. Five skaters are competing in the County Championship Finals: Miller, Diller, Hiller, Giller, and Stiller? How many different ways can they finish first, second, third, fourth, and fifth?
      ::5个滑雪运动员在州冠军决赛中竞争:米勒、迪勒、希勒、吉尔和斯蒂耶尔?他们能用多少种不同的方式完成第一、第二、第三、第四和第五场决赛?

    Find the number of permutations for each problem.
    ::查找每个问题的变换次数 。

    1. How many 3-letter words can Brenda write using the letters A, B, and C without repeating any of the letters?
      ::布伦达能用A、B和C字母写多少个3个字母的字 而不重复这些字母?
    2. How many 4-digit numbers can Brenda write using the digits 1, 3, 5, 7 without repeating any of the digits?
      ::布伦达能用数字1、3、5、7 来写多少四位数数字而不重复数字?
    3. Doug, Eileen, Francesca, and Garth all entered the swimming race. In how many different orders can the four racers finish?
      ::道格,艾琳,弗朗切斯卡,加思 都参加了游泳赛
    4. Miguel is serving soup, salad, pasta, and fish for dinner. In how many different orders can he serve the four dishes?
      ::Miguel在做汤、沙拉、意大利面和鱼做晚饭
    5. Mike has 4 different playing cards: Ace, King, Jack, and Ten. How many different 4-card arrangements can he make?
      ::麦克有4张不同的扑克牌:Ace、King、Jack和10张。他能做多少不同的4张牌安排?
    6. Marlena strung 5 charms on a bracelet–a star, a fish, a diamond, a moon, and a baby shoe. Into how many different orders can she arrange the 5 charms?
      ::玛莲娜在手镯上挂了5个护身符 — — 一颗星星、一条鱼、一颗钻石、一颗月亮和一只婴儿鞋。 她能安排多少不同顺序的5个护身符?
    7. Michelle forgot her 6-letter computer password. She knows she used the letters H, I, J, K, L, and M in the password and that she didn’t repeat any of the letters. How many different passwords must she try before she is sure to hit the correct one?
      ::米歇尔忘了她的6个字母的计算机密码。 她知道她用H、I、J、K、L和M等字母在密码中,而且她没有重复这些字母。 在她确定打正确密码之前,她必须尝试多少个不同的密码?
    8. Seven skaters are competing in the County Championship Finals. In how many different orders can they finish first, second, third, fourth, fifth, sixth, and seventh?
      ::7个滑雪运动员在州冠军决赛比赛中竞争。他们能完成多少不同的订单?第一、第二、第三、第四、第五、第六和第七份?
    9. If there are eight skaters competing, how does this change the number of permutations?
      ::如果有八名滑冰运动员参加比赛,这如何改变变相数目?

    Review (Answers)
    ::回顾(答复)

    Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
    ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

    Resources
    ::资源