确认组合

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Joanne, Laura and their four children are going to the beach for the day. They are packing lunches and want to choose two snacks from the five different types of snacks they have in their kitchens. Both women have oranges, pretzels, raisons, crackers, and apples available to choose from. How many combinations of the five snacks, taken two at a time, do the women have to choose from?
::Joanne、Laura和他们的四个孩子都要去海滩吃一天。她们正在打包午餐,想从厨房里五种不同的零食中选择两份零食。这两个妇女都有橙子、脆饼、存在的理由、饼干和苹果可供选择。五份零食的组合,一次吃两份,这些妇女必须做出选择吗?

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In this concept, you will learn to recognize combinations.
::在这个概念中,你会学会认识组合。

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Combination
::合并

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Order is important for some groups of items but not important for others. If you are making a salad and you put in some lettuce, carrot, cucumber, and green pepper, then the order in which they are put in the bowl does not natter. Hence you are talking about a . For combinations, you are merely selecting. A combination is an arrangement of items in which order, or how the items are arranged, is not important.
::秩序对某些组项很重要, 但对其它组项并不重要 。 如果您在做沙拉, 并放一些生菜、 胡萝卜、 黄瓜和绿胡椒, 那么它们放进碗里的顺序就不会消散。 因此, 您是在说一个 。 对于组合, 您只是选择。 组合是按顺序排列的项目, 或者如何排列项目, 并不重要 。

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For combinations, the collection of one order of the items is not functionally different than any other order. Think about a pizza. It doesn’t matter which order you put on the toppings once they are all on there. You can put a combination of toppings on a pizza.
::对于组合,收集一个项目顺序在功能上与其它任何顺序没有不同。 想想比萨饼。 当它们都在上面的时候, 你点哪个顺序并不重要。 您可以在比萨饼上加一个组合。

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Sometimes order does matter like opening the combination lock on your locker. When the order does matter, then you would use a .
::有时候顺序很重要,比如打开你储物柜的组合锁。当订单确实重要时,你就会使用 。

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Let’s look at an example.
::让我们举个例子。

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Six people–Larry, Sherry, Terri, Carrie, Mary, and Harry all want to ride in a rowboat that can hold only 4 passengers. How many different groups of 4 passengers can ride in the boat?
::6个人 — — 拉里、雪利、泰里、卡丽、玛丽和哈里都想坐一条只能载4名乘客的划艇。 船上有多少4人可以搭乘不同的四人组?

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

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Larry, Sherry, Terri, Harry
::拉里、雪利、泰里、哈里

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Next, rearrange the order. Did changing the order of the items change the outcome ? If so, then order matters.
::下一步重新排列顺序。 更改项目顺序是否改变了结果? 如果是的话, 那么顺序很重要 。 @ info: whatsthis

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Sherry, Harry, Larry, Terri
::雪利、哈里、拉里、泰里

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This is a different order but the same 4 passengers.
::这是一个不同的顺序,但相同的4名乘客。

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Order is not important and you would use combinations to solve for the number of groups.
::顺序并不重要, 您会使用组合来解决组数 。

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When solving problems, combinations can be used to solve problems when order is not important. One way to find the number of combinations is to use a tree diagram .
::当解决问题时,当顺序不重要时,组合可以用来解决问题。 找到组合数的方法之一是使用树形图。

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Let’s look at an example.
::让我们举个例子。

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For his top tennis doubles team, Coach Yin is considering 3 players: Joyce, Rose, and Nica. How many different doubles teams can the coach consider?
::Yin教练的顶尖网球双人球队考虑的是3个球员:Joyce、Rose和Nica。教练能考虑多少个不同的双人球队?

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First, draw a tree diagram to show all 6 permutations of the 3 players. Order doesn’t matter in this problem.
::首先,绘制一棵树图以显示这3个玩家的所有 6 个变体。 秩序在这个问题中并不重要 。

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For example, the team of Joyce-Rose is no different than the team of Rose-Joyce.
::例如,Joyce-Rose-Rose团队与Rose-Joyce团队没有什么不同。

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So in the second tree diagram we cross out all outcomes that are repeats. This leaves 3 combinations that are not repeats.
::所以在第二个树形图中,我们横跨所有重复的结果。这留下了三种组合,它们不重复。

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Joyce-Rose, Joyce-Nica, Rose-Nica
::乔伊丝露丝 乔伊丝妮卡 玫瑰妮卡

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The answer is that there are three possible combinations.
::答案是,有三种可能的组合。

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This method of making a tree diagram and crossing out repeats is reliable, but it is not the only way to find combinations.
::这种绘制树图和跨出重复数字的方法是可靠的,但这不是找到组合的唯一办法。

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Examples
::实例

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Example 1
::例1

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Earlier, you were given a problem about the  moms packing snacks for the beach.
::早些时候,你得到一个问题 关于妈妈打包零食 在海滩上。

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Joanne and Laura are getting ready for a day at the beach. The women are packing the lunches to take with them and want to choose two of the five snacks they have available to put with the lunches.
::Joanne和Laura正在海滩上准备一天的午餐 女人们正在收拾午餐带她们一起吃

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First, list the 5 snacks.
::首先,列出5个零食清单。

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$\begin{array}{r}\text{Oranges}(O),\text{Pretzels}(P),\text{Raison}(R),\text{Crackers}(C),\text{Apples}(A)\end{array}$  
::橙色( O) , Pretzels( P) , Raison( R) , Crackers( C) , Apples( A)

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Next, draw a diagram showing all of the different combinations.
::接下来绘制一个图表,显示所有不同的组合。

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The answer is 10.
::答案是10个

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There are 10 possible combinations of choosing 2 snacks from the 5 choices.
::从5种选择中选择2种零食,有10种可能的组合。

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Example 2
::例2

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How many different violin duos can Ben, Jen, Ren, Wen, and Ken form?
::Ben, Jen, Ren, Wen, 和 Ken能组成多少个不同的小提琴?

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First, list all of the possible combinations.
::首先,列出所有可能的组合。

	Ben	Jen	Ren	Wen	Ken
Ben		Ben – Jen	Ben – Ren	Ben – Wen	Ben – Ken
Jen	Jen – Ben		Jen – Ren	Jen – Wen	Jen – Ken
Ren	Ren – Ben	Ren – Jen		Ren – Wen	Ren – Ken
Wen	Wen – Ben	Wen – Jen	Wen – Ren		Wen – Ken
Ken	Ken – Ben	Ken – Jen	Ken – Ren	Ken – Wen

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Second, cross out all duplicates. For example Jen – Ben is the same as Ben – Jen.
::其次,将所有复制品都划出来。 比如,仁—本和本—仁。

	Ben	Jen	Ren	Wen	Ken
Ben		Ben  Jen	Ben  Ren	Ben  Wen	Ben  Ken
Jen	Jen – Ben		Jen  Ren	Jen  Wen	Jen  Ken
Ren	Ren – Ben	Ren – Jen		Ren  Wen	Ren  Ken
Wen	Wen – Ben	Wen – Jen	Wen – Ren		Wen  Ken
Ken	Ken – Ben	Ken – Jen	Ken – Ren	Ken – Wen

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Then, list the possible combinations of violin duos.
::然后列出小提琴的组合

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Jen – Ben
::简 - 本

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Ren – Ben
::本 -本

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Wen – Ben
::文文本

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Ken – Ben
::肯 - 本

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Ren – Jen
::任 - 简

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Wen – Jen
::文简

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Ken – Jen
::肯 - 简

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Wen – Ren
::温 - 廉

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Ken – Ren
::肯 - 任

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Ken – Wen
::肯 - 文文

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The answer is 10.
::答案是10个

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There are 10 possible combinations for these 5 people to be placed in a violin duo.
::这5个人有10种可能的组合 被放在小提琴二人组里

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Example 3
::例3

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How many different letter arrangements can you make if you have six letters but only use three at a time?
::如果你有6个字母,但每次只使用3个字母,你能做出多少不同的字母安排?

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First, choose 6 letters.
::首先,选择6个字母。

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A, B, C, D, E, F
::A、B、C、D、E、F

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Next, choose one letter and draw a diagram showing all of the different combinations.
::接下来,选择一个字母,绘制图表,显示所有不同的组合。

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The answer is that there are 20 combinations.
::答案是,有20个组合。

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Example 4
::例4

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How many different letter arrangements can you make if you have six letters, but only use four at a time?
::如果你有六个字母,但每次只使用四个字母,你能做出多少不同的字母安排?

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First, choose 6 letters.
::首先,选择6个字母。

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A, B, C, D, E, F
::A、B、C、D、E、F

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Next, choose one letter and draw a diagram showing all of the different combinations.
::接下来,选择一个字母,绘制图表,显示所有不同的组合。

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The answer is that there are 15 combinations.
::答案是,有15个组合。

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Example 5
::例5

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You have 5 colored pens. How many different letter arrangements can you make if you only choose two at a time?
::您有 5 支彩色笔。 如果您每次只选择两支, 您可以做多少个不同的字母安排 ?

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First, choose 5 colors.
::首先,选择五色。

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$\begin{array}{r}\text{Red}(R),\text{Blue}(B),\text{Black}(Bl),\text{Green}(G),\text{Purple}(P)\end{array}$
::红 (R)、 蓝 (B)、 黑 (Bl)、 绿 (G)、 紫 (P)

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Next, draw a diagram showing all of the different combinations.
::接下来绘制一个图表,显示所有不同的组合。

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The answer is 10.
::答案是10个

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There are 10 possible combinations of choosing 2 pens from 5.
::从5个中选择两支钢笔有10种可能的组合。

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Review
::回顾

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Solve each combination.
::解决每一个组合。

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1. At Dudley’s Dude Ranch there are 6 riders but only 4 horses. How many different ways can a group of 4 go out on ride?
::1. 在杜德利的酷哥牧场,有6名骑手,但只有4匹马,四匹马可以走多少种不同的方式?

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2. With 4 laps to go, Dale Earnhardt Jr., Robbie Gordon, Kyle Busch, and Kasey Kahne are all in contention to win a NASCAR race. In how many different ways can the top three drivers finish?
::2. 戴尔·欧恩哈特、罗比·戈登、凯尔·布希和卡西·卡恩在赢得NASCAR比赛方面都争论不休,前三名驾驶员能用多少种不同的方式完成比赛?

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3. Ace, King, Queen, Jack, Ten, and Nine of Clubs are face down on a table. How many different 3-card hands can you draw all at once?
::3. A、K、Q、Jack、10和9个俱乐部在桌子上脸朝下,你们能同时画出多少个不同的三张牌手?

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4. 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?
::4. 袋内有4个大理石:红色、蓝色、黄色和绿色;用多少种不同的方式,你可以伸进袋内,同时抽出2个大理石,扔进杯子里?

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5. How many different 4-horn bands can you choose from a class of 10 horn players?
::5. 在10个角播放器的班级中,你能选择多少个不同的四角乐队?

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6. Eight candidates are running in the primary elections for president. How many president and vice president pairs are possible?
::6. 八名候选人正在竞选总统的初选,有多少总统和副总统可以配对?

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7. Fifteen students compete in the Geography Bee. How many different ways can three people be chosen as winners?
::7. 十五名学生在地理蜜蜂比赛,选择三个人以多少种不同方式获胜?

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8. Nine people want to ride on the banana boat but there are only 4 life jackets. How many different groups can ride on the banana boat at one time?
::8. 9个人想乘香蕉船,但只有4件救生衣,一次可以骑多少个不同的群体?

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9. The 5 last people at a movie must compete for the last 3 empty seats. How many different groups of 3 can sit and watch the movie?
::9. 电影最后5人必须竞争最后3个空席位,有多少3个不同的群体可以坐下来看电影?

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Use what you have learned to figure out combinations.
::利用你学到的东西找出组合

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10. Leah collected 3 different flowers for a bouquet – a rose, a tulip, and a daffodil. How many 2-flower bouquets can she make?
::10. 莉亚为花束收集了三朵不同的花朵 — — 一朵玫瑰、一朵郁金香和一朵花朵。 她能做多少两朵花花花花花?

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11. Leah added a lily to her flowers. How many 2-flower bouquets can she make out of a rose, a tulip, a daffodil, and a lily?
::11. 莉亚在花朵上加了一朵百合花,她能用玫瑰、郁金香、水仙花和百合花做多少两花花花束?

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12. How many 3-flower bouquets can Leah make out of a rose, a tulip, a daffodil, and a lily?
::12. 莉亚能用玫瑰、郁金香、水仙和莉莉做多少三花花花束?

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13. How many 2-flower bouquets can Leah make out of a rose, a tulip, a daffodil, a lily, and a violet?
::13. 莉亚能用玫瑰、郁金香、水仙、百合花和紫罗兰做多少两花花花束?

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14. How many 3-flower bouquets can Leah make out of a rose, a tulip, a Daffodil, a lily, and a violet?
::14. 莉亚能用玫瑰、郁金香、多萝卜、百合花和紫罗兰做多少三花花花束?

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15. At Dudley’s Dude Ranch there are 5 dudes who want to ride – Peg, Greg, Meg, Sue, and Drew – but only 4 horses. How many different 4-horse groups can go out for a ride?
::15. 在杜德利的酷哥牧场,有5个人想骑马 — — Peg、Greg、Meg、Sue和Drew — — 但只有4匹马。 有多少不同的4匹马可以出去兜风?

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Review (Answers)
::回顾(答复)

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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 。

 

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Resources
::资源