Section outline

  • Combinations and Permutations 
    ::组合和变式

    Combinations and Permutations are each different methods of counting the possible number of ways that the members of a set may be selected or arranged. The difference between the two is that permutations consider the order in which objects are placed to be important, and combinations only consider which objects are chosen, ignoring the order.
    ::组合和变式是计算一组成员可能选择或安排的方式的每一种不同方法。两种方法的区别在于,变式考虑对象放置的重要顺序,组合只考虑选择了哪些对象,而忽略了顺序。

    lesson content

    One way to remember the difference between the two is to consider the meanings of the root words: combine and permute. You may not be familiar with the word ‘permute’, which basically means to mix up the order of something, but you are likely quite comfortable with ‘combine’, which means group things together.
    ::记住两者区别的一个方法是考虑根词的含义:组合和刻度。你可能不熟悉“permute”一词,这个词基本上意味着混淆某事物的顺序,但你可能对“combine”相当满意,这意味着将事物组合在一起。

    • Permutations: If you are trying to find the number of ways that different items can be put in order, meaning that 1, 2, 3 is one , and 3, 2, 1 is another permutation, each counting as separate entries, you are identifying the count of the different ways you could permutate or permute  (mix up) the items. It is quite possible to have multiple permutations using exactly the same items.
      ::变式 : 如果您正在试图找到不同项目可以排列的多个方法, 意思是 1, 2, 3 是一, 3, 2, 1 是另一种变式, 每个都计为单独的条目, 您正在识别您可以调和或调整( 混合) 项目的不同方法的计数 。 使用完全相同的项目可以有多个变式 。
    • Combinations: If you are identifying the number of different groups of items, meaning that 1, 2, 3 and 3, 2, 1 are the same group, regardless of order, then you are counting the number of ways you could combine (group up) the items. Combinations can also be thought of as putting items in a bucket, the order in which you drop the items in does not change which items are in the bucket when you finish. In order to have different combinations, you need different items , you can’t just drop them into the bucket in a different order.
      ::组合 : 如果您正在识别不同组别的项目数量, 这意味着 1, 2, 3, 3, 3, 3, 3, 2, 1 是相同的组, 不论顺序如何, 那么您正在计算您可以合并( 组合) 项目的方式的数量 。 组合也可以被视为将项目放进桶中, 将项目放入桶中的顺序不会改变您完成时项目在桶中的位置 。 为了使用不同的组合, 您需要不同的项目, 您不能按不同的顺序把它们放入桶中 。

    Understanding the Difference Between Combinations and Permutations 
    ::了解组合和变异之间的差异

    1. Suppose you have 12 tabs open in your web browser, and you are curious how many ways they could be reorganized. Is this a problem, or a permutation problem, and why?
    ::1. 假设你的网络浏览器中打开了12个标签,你很想知道它们可以重组多少种方式。这是问题还是变换问题,原因何在?

    lesson content

    Since you are counting the number of different ways the same 12 tabs could be organized or put in order, this is a permutation problem. Each different permutation will have the same twelve tabs, they will just be in a different order.
    ::由于您在计算不同方式的数量, 相同的 12 个标签可以按顺序组织或设置, 这是一个变异问题 。 每个不同的变异将具有相同的 12 个标签, 它们将处于不同的顺序 。

    2. Your sports team is planning a trip to a nearby town, about 2 hours away by bus. The driver will allow each rider to bring 2 electronic devices for the trip. You have access to a smartphone, a tablet, an mp3 player, and a laptop computer, and you are curious how many options that allows you to choose. Is this a combination problem, or a permutation problem, and why?
    ::2. 运动队计划乘公交车前往附近的城镇,大约2小时,司机将允许每位骑手携带2台电子设备,您可使用智能手机、平板电脑、平板电脑、平板电脑、笔记本电脑和笔记本电脑,并想知道有多少选项允许您选择。这是综合问题还是变异问题,原因何在?

    Regardless of which item you choose first, and which you choose second, the same two items will result in the same devices available to you for the trip, so order does not matter. This is a combinations problem.
    ::无论您首先选择哪个项目, 并且选择第二个项目, 同样的两个项目将产生与您在旅行中可用的设备相同的设备, 所以顺序并不重要 。 这是一个组合问题 。

    3. In the United States, telephone numbers are numerical series composed of a 3-digit area code, a 3-digit exchange, and a 4-digit subscriber line identifier. For each of the over 250 different area codes, there are 7,920,000 useable phone numbers (some numbers, particularly those starting with 0 or 1, are unavailable for public use). Is the count of useable phone numbers a product of a permutation calculation, or a combination calculation, and why?
    ::3. 在美国,电话号码是数字序列,由3位数区域代码、3位数交换机和4位数用户线路标识码组成,在250多个不同区域代码中,每个代码都有7 920 000个可用电话号码(有些数字,特别是从0或1开始的号码,无法公开使用)。

    Phone numbers are a permutation function, since 555-1234 would ring a different person than 555-4321.
    ::电话号码是一种变换功能,因为555-1234电话会按与555-421电话不同的人键。

    Earlier Problem Revisited
    ::重审先前的问题

    Evan, his girlfriend, and 5 of his other friends are going to the movies. Evan wants to be sure that he sits next to his girlfriend, and is curious how many different ways he and his friends can be arranged to sit in the same row. Is this a combination problem or a permutation problem? How can you tell?
    ::埃文,他的女朋友,还有他的其他五个朋友要去看电影。埃文希望确保他坐在他女朋友旁边,并想知道他和他的朋友能用多少种不同的方式坐在同一排里。这是综合问题还是变相问题?你怎么知道呢?

    This is a permutation problem, since Evan is interested in learning how many different ways he could order the same 7 friends. If he is curious about how many different groups of 3 friends could be chosen to go get drinks and popcorn after they have seats, then he would be dealing with a combination problem.
    ::这是一个变迁问题,因为埃文想知道他能用多少种不同的方式给同样的7个朋友点餐。 如果他想知道有多少不同的3个朋友群体在座位后可以选择去买饮料和爆米花,那么他就会面对一个综合问题。

    Examples 
    ::实例

    Example 1
    ::例1

    The back of the box of jellybeans that I have says that there are hundreds of different flavor combinations possible by eating multiple beans at the same time. If I wanted to calculate how many possibilities there were if eating only two beans at a time, would that be a permutation calculation, or a combination calculation?
    ::果冻豆盒的背面,我说过,同时吃多种豆子可以同时使用数百种不同的口味组合。 如果我想计算一次只吃两只豆子的可能性,那么这是一种变换计算,还是一种组合计算?

    Since two jellybeans of given flavors will result in the same combined flavor regardless of which I grab from the box first, this is a combination problem.
    ::由于两种特定口味的果冻 将产生相同的混合口味 不管我先从盒子里抢来哪种口味 这都是个综合问题

    Example 2
    ::例2

    Tuscany has 500 songs in her MP3 collection, and she wants to make a playlist for her boyfriend. The trick is that she wants to hide a secret message in the playlist by choosing songs so that the first word of each song forms a message. If she wanted to know how many different messages were possible from a playlist 15 songs long, including messages that are gibberish, is this a permutation or combination problem?
    ::托斯卡纳的MP3收藏中有500首歌曲,她想为她的男朋友制作一个播放列表。 关键在于她想通过选择歌曲在播放列表中隐藏一个秘密信息,让每首歌曲的第一个单词形成一个信息。 如果她想知道播放列表的15首长长的歌曲中有多少不同信息是可能的,包括胡言乱语的信息,这是否是一个变相或组合的问题?

    Since the message would be different if the same words were read in a different order, this is a permutation problem.
    ::由于如果以不同顺序阅读相同的词句,信息就会不同,因此这是一个变换问题。

    Example 3
    ::例3

    Scott has 5 dogs, and he feeds them one at a time so they don’t fight over the food. If he wants to discover how many different ways he could choose to order them for feeding, is he considering permutations or combinations?
    ::斯科特有五只狗,他一次喂一只,这样他们就不会为食物而争吵。 如果他想找出他可以选择多少种不同的方式来命令他们喂食,他考虑的是变异或组合吗?

    The order of feeding is all the Scott is considering, so this is a permutation problem. The combination of 5 dogs is the same regardless.
    ::史考特只考虑喂食顺序,所以这是一个变异问题。五只狗的组合是相同的,不管如何。

    Example 4
    ::例4

    Vicki is making gift bags for a party. She has 8 kinds of gifts and the bags hold 3 gifts each. If she wants to know how many different gift bags she can make, is she dealing with permutations or combinations?
    ::Vicki在为派对做礼品袋,她有8种礼物,每袋有3个礼物。如果她想知道她能做多少个不同的礼品袋,她处理的是变相或组合吗?

    If Vicki chooses a particular group of 3 gifts for a bag, that bag will end up the same regardless of which of the 3 she chooses first, second, or third. This is a combination problem.
    ::如果Vicki选择了一组三件礼物作为袋子,那么这个袋子最终会变成相同的,不管她选择哪一个,哪一个,第二个,还是第三个。这是一个综合问题。

    Review 
    ::回顾

    For problems 1 – 14, identify each situation as either a permutation or a combination consideration:
    ::对于问题1-14,将每一种情况确定为一种变换或综合考虑:

    1. An ice cream store has fifteen different flavors of ice cream, you wonder how many different three-scoop bowls can be made.
      ::一家冰淇淋店有15种不同的冰淇淋口味, 你想知道有多少不同的三勺碗可以做。
    2. There is a red, a blue, a green, and a yellow chair around the table. You wonder how many ways can four friends sit around the table.
      ::桌子周围有红色、蓝色、绿色和黄色的椅子。你不知道四位朋友能坐在桌子上坐几条路。
    3. Two friends decide to stay home on a Saturday and watch a movie marathon. Of the twenty movies the friends have to choose from, each friend chooses a first, second, and third choice. How many different six-movie marathons are possible?
      ::两个朋友决定周六留在家里看电影马拉松。 在朋友必须选择的20部电影中,每个朋友选择第一、第二和第三个选择。 有多少不同的6部马拉松可以办到?
    4. There are seven rides at the carnival, but you can only afford to ride four of them, how many different groups of four rides are there?
      ::嘉年华有七次骑车 但你只能骑四次 有多少组四人骑车?
    5. How many five-letter groups can you make from the word “grandmother”?
      ::你能用“祖母”这个词做成多少五个字母组?
    6. Fourteen friends decide to visit each other’s houses to trick-or-treat, but they only have time to choose seven of the houses. How many ways could the friends put the houses in order for visiting?
      ::14位朋友决定互访对方的房屋,以进行欺骗或治疗,但他们只有时间选择7栋房屋。 朋友们可以用多少方式安排这些房屋进行探访?
    7. How many different pizzas can be made, assuming no double toppings, from seven toppings?
      ::可以做多少个不同的比萨饼? 假设从7个比萨饼到7个比萨饼 不会有双倍的比萨饼吗?
    8. How many 5-card hands are possible with 20 different cards?
      ::用20张不同的卡片 有多少5张纸牌的手是可能的?
    9. You are responsible for selecting a lead and an understudy for a school play.
      ::您负责为学校剧本选择首选和替补。
    10. You are team captain for a tug-of-war, how many different teams could you create from a pool of 30 players?
      ::你们是拖战队队长 你们能从30个球员中 创造出多少不同的球队?
    11. Scott is a safety-conscious rider, and knows he should wear a leather jacket and helmet when he rides a motorcycle. There are 10 different helmets and 7 different jackets to choose from, he wonders many options are possible.
      ::斯科特是一个有安全感的骑手,他知道骑摩托车时应该穿皮革夹克和头盔。 10件不同的头盔和7件不同的夹克可以选择,他想知道许多选择是可能的。
    12. John is practicing a card trick to show his granddaughter, he asks her to pick three cards from the deck and put them back into the pile. John wants to know many groups of three he needs to guess from to get it right.
      ::约翰在练习玩牌把戏给孙女看,他要求她从甲板上挑三张牌,然后把它们放回堆里。约翰想知道他需要猜到多少组三人才能把牌弄好。
    13. Robin is kind of a health food fanatic, and she is making whole wheat pizza with organic sauce, for dinner. She has spinach, kale, fresh tomato, organic anchovy, free-range chicken, and grass-fed beef. Her kids want to know how many different pizzas she could make.
      ::Robin是个健康食品狂热主义者,她正在做全麦小麦披萨,配上有机酱,做晚饭。她有菠菜、甘蓝、新鲜番茄、有机、免费鸡肉和草食牛肉。她的孩子想知道她能做多少不同的披萨。

    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 。