12.1 适用基本计数原则

章节大纲

• 选择节常规

  折叠 展开

  常规

  全部折叠 展开全部

  • 选择活动新闻通告

    

    新闻通告 讨论区
• 选择节适用基本计数原则

  折叠 展开

  适用基本计数原则

  播放段落

  A veteran's license plate consists of digit (0-9) followed by a capital letter (A-Z) followed by a second digit (0-9). How many possible license plate combinations are there?
  ::退伍军人的牌照由数字(0-9)组成,后面是大写字母(A-Z),后面是第二个数字(0-9)。 有多少可能的牌照组合?

  播放段落

  Fundamental Counting Principle
  ::基本计数原则

  播放段落

  Sometimes we want to know how many different combinations can be made of a variety of items. The  states that the number of ways in which multiple events can occur can be determined by multiplying the number of possible outcomes for each event together. In other words, if events $\begin{array}{r}A\end{array}$ , $\begin{array}{r}B\end{array}$ , and $\begin{array}{r}C\end{array}$ have 5, 3 and 4 possible outcomes respectively, then the possible combinations of outcomes is $\begin{array}{r}5\times 3\times 4=60\end{array}$ .
  ::有时,我们想知道对各种各样的项目可以做出多少不同的组合。 声明指出, 发生多重事件的方式可以由每个事件的可能结果的乘数一起来决定。 换句话说, 如果事件A、 B和 C分别有5、 3和 4的结果, 那么结果的可能组合是 5x3x4=60 。

  播放段落

  Let's solve the following problems.
  ::让我们解决以下的问题。

  播放段落
  1. Sofia works in a clothing store. She has been given the task of setting up a mannequin with a skirt, a shirt and a pair of shoes from a display of coordinating skirts, shirts and shoes. Since they all coordinate she can pick any shirt, any skirt and any pair of shoes and the outfit will work. If there are 3 skirts, 5 shirts and 2 pairs of shoes, how many ways can she dress the mannequin?
     ::Sofia在一家服装店工作,她的任务是用裙子、衬衫和一双鞋来装饰一个模特,用裙子、衬衫和一双鞋来装饰裙子、衬衫和鞋子,由于她们都协调,她可以挑任何衬衫、裙子、鞋子和服装。如果有3件裙子、5件衬衫和2双鞋,她可以穿多少衣服?

  播放段落

  Let’s use a tree to help us visualize the possibilities. If we start with Shirt A, we get the following possibilities for the remainder of the outfit:
  ::让我们用一棵树来帮助我们想象这些可能性。 如果我们从Shirt A开始,我们就能为其余的服装找到以下可能性:

  

  播放段落

  So we could have the following 6 combinations with Shirt A:
  ::因此,我们可以有以下6个组合 与Shirt A:

  播放段落

  Shirt A, Skirt A, Shoe A
  ::衬衫A,衬衫A,鞋A,鞋A

  播放段落

  Shirt A, Skirt A, Shoe B
  ::衬衫A、衬衫A、鞋子B

  播放段落

  Shirt A, Skirt B, Shoe A
  ::衬衫A、衬衫B、鞋子A

  播放段落

  Shirt A, Skirt B, Shoe B
  ::衬衫A、衬衫B、鞋子B

  播放段落

  Shirt A, Skirt C, Shoe A
  ::衬衫A、衬衫C、鞋A

  播放段落

  Shirt A, Skirt C, Shoe B
  ::衬衫A、衬衫C、鞋子B

  播放段落

  Consider that there are four other shirts that will also have 6 combinations of skirts and shirts that will go with them. Now, there are $\begin{array}{r}5\times 6\end{array}$ total combinations which is 30 ways that Sofia could dress the mannequin.
  ::想想还有其他四件衬衫,这些衬衫还有六件裙子和衬衫的组合。 现在,共有5×6个组合,这是索菲亚可以穿假发的30种方式。

  播放段落
  2. Ralph is trying to purchase a new car. The salesperson tells him that there are 8 different possible interior colors, 5 exterior colors and 3 car models to choose from. How many different unique cars does he have to choose from?
     ::销售员告诉他,有8种不同的内部颜色、5种外部颜色和3种汽车模型可供选择。他需要选择多少辆不同的汽车?

  播放段落

  Instead of making a tree diagram this time, let’s look at a more efficient method for determining the number of combinations. If we consider what happens in the tree diagram, the 8 different interior colors would each be matched with each of the 5 exterior colors and those combinations would then be linked to the 3 different models, we can see that:
  ::与其这次绘制树图,不如看看一个更有效的方法来决定组合数。 如果我们考虑一下树图中发生的情况,8种不同的内部颜色将分别与5种外部颜色中的每一种颜色匹配,然后这些组合将与3种不同的模式连接起来,我们可以看到:

  播放段落

  $$\begin{array}{r}8\text{interior colors}\times 5\text{exterior colors}\times \text{3 models}=8\times 5\times 3=120\text{combinations}\end{array}$$

  
  ::8 内部颜色x5 外部颜色x 3 型号= 8x5x3= 120 组合

  播放段落
  3. Monique is having a 5 course dinner in the dining room on a cruise. The menu consists of 2 appetizers, 3 soups, 2 salads, 4 entrees and 3 desserts. How many different meals could be configured if she chooses one of each course?
     ::Monique在餐厅吃五道菜。菜单包括2个开胃菜、3个汤、2个沙拉、4个主菜和3个甜点。如果她每道菜选一餐,可以配置多少餐?

  播放段落

  Following the method described in #2 above, we can multiply the number of choices for each course together to determine the total combinations:
  ::按照上文#2中描述的方法,我们可以乘以每一课程的选择数,以确定总组合:

  播放段落

  $\begin{array}{r}2\times 3\times 2\times 4\times 3=144\text{unique}5\text{course meals}\end{array}$
  ::2x3x2x2x4x3=144 独特的5个课程餐

  播放段落

  Examples
  ::实例

  播放段落

  Example 1
  ::例1

  播放段落

  Earlier, you were asked to find the total number of  possible license plate combinations.
  ::早些时候,你被要求找到 可能牌照组合的车牌总数

  播放段落

  We can multiply the number of choices for each license plate position together to determine the total combinations:
  ::我们可以乘以每个牌照牌照位置的选择数,以确定总组合:

  $\begin{array}{r}10\times 26\times 10=2600\end{array}$

  播放段落

  Therefore, there are 2600 possible license plate combinations.
  ::因此,有2 600个可能的牌照组合。

  播放段落

  Example 2
  ::例2

  播放段落

  A coffee shop offers a special espresso deal. You choose one of three sizes, one of 5 flavored syrups and whole, nonfat or soy milk. How many drink combinations can be made?
  ::咖啡店提供一种特别的咖啡交易。您可以选择三种体型之一,五口味糖浆之一和全脂、非脂肪或大豆牛奶。可以做多少饮料组合?

  播放段落

  There are 3 sizes, 5 syrups and 3 kinds of milk from which to choose. So, there are  $\begin{array}{r}3\times 5\times 3=45\text{drinks}\end{array}$ .
  ::有3个尺寸,5个糖浆和3种牛奶可供选择。所以有3x5x3=45的饮料。

  播放段落

  Example 3
  ::例3

  播放段落

  Sarah goes to a local deli which offers a soup, salad and sandwich lunch. There are 3 soups, 3 salads and 6 sandwiches from which to choose. How many different lunches can be formed?
  ::Sarah去一个当地熟食店,提供汤、沙拉和三明治午餐。有3个汤、3个沙拉和6个三明治可供选择。可以组成多少个不同的午餐?

  播放段落

  $\begin{array}{r}3\text{soups}\times 3\text{salads}\times 6\text{sandwiches}=54\text{lunch combos}\end{array}$
  ::3个汤x3 沙拉x6 三明治=54个午餐包

  播放段落

  Example 4
  ::例4

  播放段落

  A design your own t-shirt website offers 5 sizes, 8 colors and 25 designs for their shirts. How many different t-shirts can be designed?
  ::你自己的T恤网站的设计提供5尺寸、8颜色和25件衬衫设计。可以设计多少件不同的T恤?

  播放段落

  $\begin{array}{r}5\text{sizes}\times 8\text{colors}\times 25\text{designs}=1000\text{shirts}\end{array}$
  ::5 大小Sxx8 颜色x25 设计= 1 000 衬衫

  播放段落

  Review
  ::回顾

  播放段落

  Use the Fundamental Counting Principle to answer the following questions.
  ::使用基本计算原则回答下列问题。

  播放段落
  1. A frozen yogurt shop has a half price Sunday Sundae special. Customers can get one of four flavors, one of three syrups and one of twelve toppings on their sundae. How many possible sundae combinations can be made?
     ::冷冻酸奶店有半价的周日圣代特价。 顾客可以在其圣代上得到四种口味之一、三个糖浆之一和十二个奶粉之一。 有多少可能的圣代组合?
  播放段落
  2. At a neighborhood restaurant wings are the specialty. The restaurant offers 3 sizes of wings, 4 levels of heat and ranch or blue cheese dipping sauce. How many different orders are possible?
     ::餐厅提供3码的翅膀,4级的热量和牧场,或蓝奶酪酱。可以订购多少份不同的订单?
  播放段落
  3. A noodle restaurant offers five types of noodles to choose from and each dish comes with a choice of one of four meats and six different sauces. How many combinations can be made?
     ::面条餐厅提供五种面条供选择,每道菜都配有四份肉和六份不同的酱汁。可以做多少次配方?
  播放段落
  4. Charlie flips a coin and then rolls a die. How many different outcomes are possible?
     ::Charlie翻了硬币,然后就死了。可以产生多少不同的结果?
  播放段落
  5. On a one week cruise, the ship stops in four ports. At each port there are six different excursions to choose from. If a passenger chooses one excursion at each port, how many different vacation experiences can be created?
     ::在一周的巡航中,船舶停靠在四个港口,每个港口有六个不同的游览可选择。 如果乘客选择在每一个港口进行一次游览,那么可以创造多少不同的度假经验?
  播放段落
  6. Samuel wants to know if he can go a whole month without wearing the exact same outfit twice. He has three pairs of pants, six shirts and two pairs of shoes. Can he make a unique outfit for each day of the month?
     ::塞缪尔想知道他能否在不穿两件完全相同的衣服的情况下去整整一个月。他有三条裤子、六件衬衫和两双鞋子。他能为每个月每天做一套独特的衣服吗?
  播放段落
  7. A car dealership has four different models to choose from in six exterior colors. If there are three different interior colors to choose from, how many different vehicles can be designed?
     ::汽车经销商有四种不同的模式,可以选择六种外部颜色。 如果有三种不同的内部颜色可以选择,那么可以设计多少种不同的车辆?
  播放段落
  8. A burrito bar offers a lunch special burrito. Customers can choose a flour or corn tortilla; chicken, steak or carnitas; white or brown rice; black beans or pinto beans; cheese, guacamole, or sour cream; and one of four salsas for a special price. How many different burritos can be made?
     ::玉米卷饼酒吧提供特餐玉米饼。 顾客可以选择面粉或玉米玉米玉米饼;鸡肉、牛排或肉桂;白米或棕米;黑豆或扁豆;奶酪、瓜卡莫乐或酸奶油;以及四个沙司中的一个,以特别价格购买。 可以做多少不同的玉米饼?
  播放段落
  9. Maria rolls a die, spins a spinner with four numbers and then flips a coin. How many possible outcomes are there?
     ::Maria滚动死亡,用四个数字旋转一个旋转器,然后翻一个硬币。有多少可能的结果?
  播放段落
  10. A local restaurant offers a dinner special. Diners can choose one of six entrees, one of three appetizers and one of 3 desserts. How many different meals can be formed?
      ::当地餐厅提供餐饮特餐。 晚餐可以选择六种菜中的一种, 三种开胃菜中的一种, 三种甜点中的一种。 有多少不同餐饮可以组成?

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  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 。