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  • 合并

    lesson content

    Jim and Maralie are in charge of tossing bracelets out into the audience for the school float during the town's Holiday Parade.  There are a total of four colors of bracelets:  orange, red, green, and blue.  They will reach into a basket and toss out three at a time.  Is this a or a ?
    ::吉姆和玛拉莉负责把手镯扔到学校的观众中, 在镇上的假日游行中, 学校浮标。 共有四种手镯颜色: 橙色、红色、绿色和蓝色。 它们会伸向篮子, 一次扔出三只。 这是一只还是一只?

    In this concept, you will learn the difference between combinations and permutations.  
    ::在这个概念中,你会了解组合和变异之间的区别。

    Combinations
    ::合并

    A combination is a collection of items in which order, or how the items are arranged, is not important. Consider a list of three words: HOPS, SHOP, and POSH.
    ::组合是一系列项目,其顺序或项目安排方式都不重要。 考虑一个三个词的列表:HOPS、 SHOP 和 POSH 。

    • For the spelling of each individual word, order is important. The words HOPS, SHOP, and POSH all use the same letters, but spell out very different words.
      ::对于每个单词的拼写,顺序很重要。 HOPS、SHOP和POSH三个词都使用相同的字母,但用非常不同的词来表达。
    • For the list itself, order is not important. Whether the words are presented in one order–such as HOPS, SHOP, POSH, or another order, such as SHOP, POSH, HOPS, or a third order, such as POSH, HOPS, SHOP–makes no difference. As long as the list includes all 3 words, the order of the 3 words doesn’t matter.
      ::对清单本身来说,顺序并不重要。 单词是按一个顺序排列的 — — 如HOPS、SHOP、POSH等,还是按另一个顺序排列的 — — 如SHOP、POSH、HOPS等,还是按第三顺序排列的 — — 如POSH、HOPS、SHOP等 — — 并不重要。 只要清单包括所有三个单词,3个单词的顺序就无关紧要了。

    Combinations and permutations are related. To solve problems in which order matters, you use permutations . To solve problems in which order does NOT matter, use combinations .
    ::组合和排列是相互关联的。为了解决顺序问题,您使用排列。要解决顺序无关紧要的问题,您使用组合。要解决顺序无关紧要的问题,请使用组合。

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

    The winning 3-digit lottery numbers are drawn from a drum as 641, 224, and 806. Does order matter in the way the three winning numbers are drawn?
    ::3位数的中奖彩票号码是641,224和806。

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

    641, 224, 806

    Next, rearrange the order. Did you change the outcome ? If so, then order matters.
    ::接下来,重新排列顺序。 您更改了结果吗 ? 如果是的话, 那么命令很重要 。 @ info: whatsthis

    224 , 806 , 641 ⟸ different order, same 3 winning numbers
    ::224,806,641 不同的顺序, 相同的3个赢数

    Order does NOT matter for this problem. 
    ::命令对此问题无关紧要。

    Here's another example.
    ::下面是另一个例子。

    A bag has 4 marbles: red, blue, yellow, and green. In how many different ways can you reach into the bag and draw out 1 marble, then return the marble to the bag and draw out a second marble?
    ::一个袋子有4个大理石:红色、蓝色、黄色和绿色。你可以用多少种不同的方式伸进袋子,抽出1个大理石,然后把大理石还给袋子,再抽出第二块大理石?

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

    red, blue
    ::红色, 蓝色

    Next, rearrange the order. Did you change the outcome? If so, then order matters.
    ::接下来,重新排列顺序。 您更改了结果吗 ? 如果是的话, 那么命令很重要 。 @ info: whatsthis

    blue, red ⟸ different order, meaning is DIFFERENT
    ::蓝色 红色 不同顺序 意思不同

    Order DOES matter for this problem.
    ::命令DOES与这一问题有关。

    Examples
    ::实例

    Example 1
    ::例1

    Earlier, you were given a problem about Jim and Maralie's parade responsibility of tossing out bracelets from their school float during the Holiday Parade.
    ::早些时候,你被问及 Jim和Maralie的游行责任 是在节日游行期间 从学校的浮标上扔掉手镯

    They will be tossing out three bracelets at a time taken from a basket with bracelets that are either orange red, green, or blue.  Is this an example of a permutation or a combination?
    ::他们每取出三只手镯,就用橙红色、绿色或蓝色的手镯,从篮子中取出三只手镯。这是变形的例子呢?还是混合的呢?

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

    red, green, blue
    ::红色、绿色、蓝色

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

    green, red, blue
    ::绿色、红色、蓝色

    Then, evaluate whether the order does matter given the situation.
    ::然后,根据情况,评估命令是否重要。

    The answer is it does not make a difference in which color order the bracelets are tossed.  This is a combination.
    ::答案是它不会改变手镯被丢弃的颜色顺序。 这是一个组合 。

    Example 2
    ::例2

    Is the following a permutation or a combination?
    ::以下是变异还是组合?

    Mario’s gym locker uses the numbers 14, 6, and 32. How many different arrangements of the three numbers must Mario try to be sure he opens his locker?
    ::马里奥的健身房储物柜使用14号、6号和32号,

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

    6, 14, 32

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

    14, 32, 6

    Then, evaluate whether the order does matter given the situation.
    ::然后,根据情况,评估命令是否重要。

    The answer is there is only one way that the numbers can be arranged to open Mario's locker. This is a permutation because order does matter.
    ::答案是,只有一种方法可以将数字排列为打开Mario的储物柜。这是一个变换,因为秩序很重要。

    Write whether each example is a permutation or combination.
    ::写下每个示例是变换还是组合。

    Example 3
    ::例3

    Cesar the dog-walker has 5 dogs but only 3 leashes. How many different ways can Cesar take a walk with all 3 dogs at once?
    ::塞萨尔有5只狗,但只有3只皮带。

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

    Dog 1 - Leash 1
    ::狗1 - 皮带1

    Dog 2 - Leash 2
    ::狗2 - 皮带2

    Dog 3 - Leash 3
    ::狗狗 3 - Leash 3

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

    Dog 1 - Leash 2
    ::狗1 - 皮带2

    Dog 2 - Leash 1
    ::狗狗2 - 皮带1

    Dog 3 - Leash 3
    ::狗狗 3 - Leash 3

    Then, evaluate whether the order does matter given the situation.
    ::然后,根据情况,评估命令是否重要。

    The answer is the order does not matter.  The dogs will get walked on leashes no matter the order.  This is a combination.
    ::答案是命令无关紧要。不管命令是什么,狗都会被绑在皮带上。这是一个组合。

    Example 4
    ::例4

    Five different horses entered the Kentucky Derby. How many different ways can the horses finish the race?
    ::5匹不同的马进入肯塔基德比,马能用多少种不同的方式完成比赛?

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

    Horse 2, Horse 4, Horse 5
    ::马2,马4,马5,马2,马4,马5,马2,马2,马4,马5,马2,马2,马2,马2,马4,马5

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

    Horse 4, Horse 2, Horse 5
    ::马,马,马,马,二,马,五,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马,马

    Then, evaluate whether the order does matter given the situation.
    ::然后,根据情况,评估命令是否重要。

    The answer is the order does matter.  The order of the horses tells what place they come in for the race. This is a permutation.
    ::答案是顺序很重要。马的顺序说明他们来参加比赛的地点。这是一个变换。

    Example 5
    ::例5

    How many different 5-player teams can you choose from a total of 8 basketball players?
    ::你能从总共8个篮球运动员中 选出多少个不同的5人球队?

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

    Player 1, Player 3, Player 4, Player 5, Player 7
    ::玩家 1,玩家 3,玩家 4,玩家 5,玩家 7

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

    Player 4, Player 3, Player 7, Player 1, Player 5
    ::玩家4,玩家3,玩家7,玩家1,玩家5

    Then, evaluate whether the order does matter given the situation.
    ::然后,根据情况,评估命令是否重要。

    The answer is the order of choosing the teams does not matter.  This is a combination.
    ::答案是选择团队的顺序无关紧要。这是一个组合。

    Review
    ::回顾

    Write whether you are more likely to use permutations or combinations for each of the following examples.
    ::写入您是否更可能对以下每个示例使用变位或组合。

    1. A bag has 4 marbles: red, blue, yellow, and green. In how many different ways can you reach into the bag and draw out 2 marbles at once and drop them in a cup?
      ::一个袋子有4个大理石块:红色、蓝色、黄色和绿色。你能用多少种不同的方式伸进袋子,一次抽出2个大理石,然后扔进杯子里?
    2. A bag contains 5 slips of paper with letters A , B , C , D , and E written on them. Pull out one slip, mark down the letter and replace it in the bag. Do this 3 times so you have written 3 letters. How many different ways can you write the 3 letters?
      ::袋子里有5张纸条, 上面写着字母 A、 B、 C、 D 和 E 。 拿出一张纸, 记下字母并替换在袋子里 。 做三次, 这样你写了3封信。 您可以写多少种方式 ?
    3. Eight candidates are running for the 4-person Student Council. How many different Student Councils are possible?
      ::八名候选人正在竞选四人学生理事会。 有多少不同的学生理事会是可能的?
    4. Mark's gym locker uses the numbers 24, 36, and 2. How many different arrangements of the three numbers must Mark try to be sure he opens his locker?
      ::马克的健身房储物柜 使用24号、36号和2号号码 三个号码的多少不同安排 马克必须尝试确定他打开储物柜
    5. Five horn players are running for 2 seats in a jazz band. How many different ways can the two horn players be chosen?
      ::五支角乐手在爵士乐乐队中竞选两个席位。两个角乐手可以选择多少种不同的方式?

    Identify each situation as a permutation or combination.
    ::将每种情况作为变相或组合加以确定。

    1. The medals awarded for a swimming meet.
      ::游泳比赛的奖章
    2. The order in which you place toppings on a pizza.
      ::在披萨上放托普的顺序
    3. The numbers that open a combination lock.
      ::打开组合锁的号码
    4. The way that you organize your clothing in a drawer.
      ::在抽屉里整理衣服的方式
    5. A baseball line up according to ability.
      ::棒球队根据能力排成一排
    6. A group of people organized randomly.
      ::一群人随机组织起来
    7. Walking five dogs three at a time.
      ::一次走五只狗三只
    8. Walking five dogs with a specific order.
      ::步行五只狗 有特别的顺序。
    9. How many different Student Councils are possible?
      ::有多少不同的学生会是可能的?
    10. How many different Student Councils are possible?
      ::有多少不同的学生会是可能的?

    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