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    lesson content

    ::[图1]

    As chairperson for the Performing Arts Center banquet, Tabitha has to help plan the meal. The caterer gave her a choice. She could choose a grilled chicken dinner, a roast beef dinner, or she could choose the buffet, which offered a selection of three entrees (grilled chicken, roast beef, and fish) and four side dishes. Tabitha thought the buffet was the best way to go, but before she agreed to it, she wanted to know how many different meal combinations they could make from the buffet if each meal consisted of one entrée and two sides.
    ::作为表演艺术中心宴会的主席,Tabitha必须帮助规划饭菜。宴会者给了她一个选择。她可以选择一份烤鸡晚餐、一份烤牛肉晚餐,或者她可以选择一份自助餐,提供三道菜(烤鸡、烤牛肉和鱼)和四道边菜。 Tabitha认为自助餐是最好的去路,但在她同意之前,她想知道如果每餐由一面和两面组成,他们可以从自助餐中做多少不同的餐。

    In this concept, you will learn to find all possible combinations.
    ::在这个概念中,你会学会找到所有可能的组合。

    Finding Combinations
    ::查找组合

    Combinations  are used for groupings in which order does not matter.
    ::组合用于顺序无关紧要的组合。

    To figure out combinations, start by listing all of the possible options. Take a look at the following chart. You can use it to figure out all the possible 2-letter combinations you can make with the letters, A, B, C, and D.
    ::要找出组合, 请首先列出所有可能的选项 。 请查看下图 。 您可以使用它来找出您可以用字母 A、 B、 C 和 D 组合的所有可能的双字母组合 。

      A B C D
    A        
    B        
    C        
    D        

    Next, go down the chart beginning with the A column and match A with each of the letters in the rows. For example, the first column would appear like this.
    ::接下来,从 A 列开始下下图,然后将 A 与行中的每个字母匹配。例如, 第一列会像这样出现 。

      A B C D
    A AA      
    B AB      
    C AC      
    D AD    

    Next, go down the chart beginning with the A column and match A with each of the letters in the rows. For example, the first column would appear like this.
    ::接下来,从 A 列开始下下图,然后将 A 与行中的每个字母匹配。例如, 第一列会像这样出现 。

    You would do the same for each of the other columns. When you’re done, the chart would look like this.
    ::其它的栏目都一样。 当你完成后,图表会像这样。

      A B C D
    A AA BA CA DA
    B AB BB CB DB
    C AC BC CC DC
    D AD BD CD DD

    Then eliminate any duplicates. For example, AB and BA are duplicates because they are the same two letters but the order is reversed. So, BA can be eliminated.
    ::然后消除任何重复。 例如, AB 和 BA 是重复的, 因为它们是相同的两个字母, 但命令被颠倒。 所以 BA 可以被删除 。

      A B C D
    A AA BA CA DA
    B AB BB CB DB
    C AC BC CC DC
    D AD BD CD DD

    The crossed-out terms are duplicates. The number of  outcomes  left is the combinations.
    ::跨出术语是重复的,剩下的结果是组合。

    Here is a real-life example.
    ::这是真实生活中的一个实例。

    Seth, Keith, Derek and Justin want to go on the bumper cars. They can only ride in pairs. How many different paired combinations are possible given these parameters?
    ::赛斯、基斯、德里克和贾斯汀都想开保险杠车,他们只能搭对车。鉴于这些参数,有多少不同的配对组合是可能的?

    To start, list all possible options beginning with Seth.
    ::要开始, 请从 Seth 开始列出所有可能的选项 。

    Seth can ride with Keith, Derek or Justin.
    ::赛斯可以和基思、德里克或贾斯汀一起骑马

    Then do the same with Keith and the others.
    ::然后对基思和其他人也这样做。

    Keith can ride with Seth, Derek or Justin.
    ::Keith可以和Seth,Derek或者Justin一起骑马

    Derek can ride with Seth, Justin or Keith.
    ::德里克可以和赛斯 贾斯汀或基思一起骑马

    And Justin can ride with Seth, Derek or Keith.
    ::贾斯汀可以和赛斯、德里克或基思一起骑马

    Here are the possible combinations.
    ::这是可能的组合。

    SK KS DS JS
    SD KD DK JK
    SJ KJ DJ JD

    Next, cross out any duplicates. It does not matter if the letters are reversed. For example, because there is SD, you would cross out DS.
    ::下一步, 划出任何复制件。 如果这些字母被倒转, 并不重要 。 例如, 因为有SD, 您会划出 DS 。

    SK KS DS JS
    SD KD DK JK
    SJ KJ DJ JD

    There are six different pair combinations left.
    ::还有六种不同的组合组合

    Examples
    ::实例

    Example 1
    ::例1

    Earlier, you were given a problem about Tabitha and her meal decision for the Honor Society’s banquet.
    ::先前,你曾因为Tabitha和她为荣誉协会宴会所作的饭食决定 遇到问题。

    Tabitha expected about 125 people for the Performing Arts Center banquet, and she decided a buffet-style dinner as the best way to go. The caterer said there would be three entrées and four sides on the buffet. A dinner consisted of one entrée and two sides. Tabitha sat down and began figuring out how many possible dinner combinations could be made from the buffet.
    ::Tabitha预计会有125人参加表演艺术中心宴会,她决定以自助餐式晚餐为最佳方式。 餐饮商说自助餐上将有3个餐桌和4个餐桌。 晚餐由1个餐桌和2个餐桌组成。 Tabitha坐下来,开始想方设法从自助餐中做多少可能的晚餐组合。

    How can you help Tabitha find all the possible combinations?
    ::如何帮助塔比塔找到所有可能的组合?

    First, list all possible options for each entree.
    ::首先,列出每个企业的所有可能选择。

    Entrée 1 Entrée 2 Entrée 3

    Next, list the side dish combinations for each entrée. Sides are labeled A, B, C, D.
    ::下一个,列出每个entrée 的侧盘组合。 侧面被标为 A、 B、 C、 D 。

    Entrée 1 Entrée 2 Entrée 3
    AB AB AB
    AC AC AC
    AD AD AD
    BA BA BA
    BC BC BC
    BD BD BD
    CA CA CA
    CB CB CB
    CD CD CD
    DA DA DA
    DB DB DB
    DC DC DC

    Then, cross out any duplicates. Remember, order does not matter in combinations.
    ::记住,在组合中,顺序无关紧要。

    Entrée 1 Entrée 2 Entrée 3
    AB AB AB
    AC AC AC
    AD AD AD
    BA BA BA
    BC BC BC
    BD BD BD
    CA CA CA
    CB CB CB
    CD CD CD
    DA DA DA
    DB DB DB
    DC DC DC

    The crossed-out combinations are duplicates.
    ::跨界组合是重复的。

    There are 18 possible combinations left.
    ::还有18种可能的组合

    The answer is the buffet offers 18 possible meal combinations.
    ::答案是自助餐提供18种可能的餐具组合。

    Example 2
    ::例2

    How many combinations are possible in the following scenario?
    ::在以下情况下,有多少组合是可能的?

    The sixth grade class voted on colors for the school flag. The top choices were red, blue, green and yellow. The students can only choose three colors.
    ::六年级学生对国旗的颜色投了票。最高选择是红色、蓝色、绿色和黄色。学生只能选择三种颜色。

    First, list all possible options beginning with red.
    ::首先,从红色开始列出所有可能的选项 。

    Red, Blue, Green Red, Blue, Yellow Red, Green, Yellow
    Blue, Red, Green Blue, Red, Yellow Blue, Green, Yellow
    Green, Red, Blue Green, Red, Yellow Green, Blue, Yellow
    Yellow, Red, Blue Yellow, Red, Green Yellow, Blue, Green

    Next, cross out any duplicates. Remember, order does not matter in combinations.
    ::下一个, 划出任何复制件。 记住, 组合中顺序无关紧要 。

    Red, Blue, Green Red, Blue, Yellow Red, Green, Yellow
    Blue, Red, Green Blue, Red, Yellow Blue, Green, Yellow
    Green, Red, Blue Green, Red, Yellow Green, Blue, Yellow
    Yellow, Red, Blue Yellow, Red, Green Yellow, Blue, Green

    The crossed-out combinations are duplicates.
    ::跨界组合是重复的。

    There are four possible combinations left.
    ::还有四种可能的组合。

    The answer is four possible combinations.
    ::答案是四种可能的组合。

    Example 3
    ::例3

    Kyle has four different pairs of sneakers. He can only bring two pairs to camp. How many different combinations can he make?
    ::凯尔有四双不同的运动鞋 他只能带两双

    First, label each pair of sneakers.
    ::首先,给每双运动鞋贴上标签

    A, B, C, D
    ::A、B、C、D

    Next, list all possible options beginning with sneakers A.
    ::下一步,从运动鞋 A 开始列出所有可能的选项 。

    AB, AC, AD
    ::AB、AC、AD

    Then, do the same for the others.
    ::那么,对其他人也做同样的事情。

    BA, BC, BD
    ::BB、BB、BB、BD、BD

    CA, CB, CD
    ::CA、CB、CD CA、CB、CD

    DA, DB, DC
    ::DA、DB、DC

    Now, cross out the duplicates. Remember, in combinations, order does not matter.
    ::现在,把复制品划出来,记住,在组合中,顺序无关紧要。

    AB, AC, AD
    ::AB、AC、AD

    BA, BC, BD
    ::BB、BB、BB、BD、BD

    CA, CB, CD
    ::CA、CB、CD CA、CB、CD

    DA, DB, DC
    ::DA、DB、DC

    There are 6 combinations left.
    ::还有6个组合

    The answer is 6 combinations.
    ::答案是6个组合

    Example 4
    ::例4

    How many different combinations can you choose from five colors taken three at a time?
    ::从五种颜色中,您可以选择多少不同的组合, 一次三种颜色?

    First, label each of the 5 colors or use the name of the color. For simplicity, let’s use letters to denote colors.
    ::首先,对5种颜色中的每一种进行标签,或者使用颜色的名称。为了简单起见,让我们使用字母来表示颜色。

    A, B, C, D, E
    ::A、B、C、D、E

    Next, list all possible 3-colors options beginning with A.
    ::下一步,从 A 开始列出所有可能的 3 色选项 。

    ABC, ABD, ABE, ACB, ACD, ACE, ADB, ADC, ADE, AEB, AEC, AED
    ::ABC、ABD、ABE、ABB、ABB、ACB、ACD、ACE、亚银、ADC、ADE、AEB、AEC、AED

    Then, do the same for the others.
    ::那么,对其他人也做同样的事情。

    BAC, BAD, BAE, BCA, BCD, BCE, BDA, BDC, BDE, BEA, BEC, BED
    ::BAC、BAD、BAD、BAE、BAD、BAD、BCD、BCE、BDA、BDC、BDE、BDE、BEA、BEC、BED 溴化二苯醚

    CAB, CAD, CAE, CBA, CBD, CBE, CDA, CDB, CDE, CEA, CEB, CED
    ::CAB、CAD、CAD、CAE、CBA、CBB、CBB、CBE、CDA、CDB、CDB、CDE、CDE、CEA、CEB、CED、CED

    DAB, DAC, DAE, DBA, DBC, DBE, DCA, DCB, DCE, DEA, DEB, DEC
    ::DAB、发援会、DAE、DBB、DBB、DBC、DBE、DCA、DCB、DCE、DEA、DEB、DEC

    EAB, EAC, EAD, EBA, EBC, EBD, ECA, ECB, ECD, EDA, EDB, EDC
    ::EAB、EAC、EAD、EBA、EBC、EBC、EBD、EBD、ECA、ECE、ECE、ECD、EDA、EDD、EDB、EDC 欧洲经济区、EDC

    Now, cross out the duplicates. Remember, in combinations, order does not matter.
    ::现在,把复制品划出来,记住,在组合中,顺序无关紧要。

    ABC, ABD, ABE, ACB, ACD, ACE, ADB, ADC, ADE, AEB, AEC, AED
    ::ABC、ABD、ABE、ABB、ABB、ACB、ACD、ACE、亚银、ADC、ADE、AEB、AEC、AED

    BAC, BAD, BAE, BCA, BCD, BCE, BDA, BDC, BDE, BEA, BEC, BED
    ::BAC、BAD、BAD、BAE、BAD、BAD、BCD、BCE、BDA、BDC、BDE、BDE、BEA、BEC、BED 溴化二苯醚

    CAB, CAD, CAE, CBA, CBD, CBE, CDA, CDB, CDE, CEA, CEB, CED
    ::CAB、CAD、CAD、CAE、CBA、CBB、CBB、CBE、CDA、CDB、CDB、CDE、CDE、CEA、CEB、CED、CED

    DAB, DAC, DAE, DBA, DBC, DBE, DCA, DCB, DCE, DEA, DEB, DEC
    ::DAB、发援会、DAE、DBB、DBB、DBC、DBE、DCA、DCB、DCE、DEA、DEB、DEC

    EAB, EAC, EAD, EBA, EBC, EBD, ECA, ECB, ECD, EDA, EDB, EDC
    ::EAB、EAC、EAD、EBA、EBC、EBC、EBD、EBD、ECA、ECE、ECE、ECD、EDA、EDD、EDB、EDC 欧洲经济区、EDC

    There are 10 combinations left.
    ::还有10个组合

    The answer is 10 combinations.
    ::答案是10个组合

    Example 5
    ::例5

    True or false. In a  , order makes a difference.
    ::无论是真实的还是虚假的 在一个...

    The answer is false. Order does not make a difference in a combination.

    ::答案是虚假的。 秩序在组合上没有区别。

    Review
    ::回顾

    Determine all possible combinations to the scenarios below and answer any questions asked about combinations or situations.
    ::确定与以下情景的所有可能的组合,并回答就组合或情况提出的任何问题。

    1. 5 colors taken four at a time
      ::5 色, 一次4 色, 5 色, 一次4 色
    2. 6 colors taken two at a time
      ::6 色, 一次两色, 6 色
    3. 6 colors taken four at a time
      ::6 色, 一次4 色, 6 色, 一次4 色
    4. 6 colors taken three at a time
      ::6 色, 一次三色, 6 色
    5. 7 dogs walked two at a time
      ::7只狗一次走两只
    6. 7 dogs walked three at a time
      ::7只狗一次走3只
    7. 7 dogs walked four at a time
      ::7只狗一次走四只四只
    8. 7 dogs walked five at a time
      ::7只狗一次走5只
    9. 13 ice cream flavors taken two at a time.
      ::13种冰淇淋口味 一次吃两个
    10. 13 ice cream flavors taken three at a time.
      ::13个冰淇淋口味 一次吃3个
    11. 13 ice cream flavors taken four at a time.
      ::13个冰淇淋口味一次吃4个
    12. 10 children arranged in groups of five
      ::10名儿童,五人一组安排10名儿童
    13. 10 children arranged in groups of four
      ::10名儿童被安排为四人一组
    14. 10 children arranged in groups of three
      ::10名儿童分三组安排

    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