8.5 计算组合

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  计算组合

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  Suppose there are fifteen students in a classroom, and you are looking to build a team of five students for a tug-of-war.  How many different groups of five students could you create?
  ::假设一个教室里有15个学生,而你正在寻找建立一个由5个学生组成的团队来进行一场拔河战争。你能创造多少个由5个学生组成的不同群体呢?

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  After the following lesson, you should have no problem solving question of this type.
  ::在以下课后,您不应遇到任何解决这类问题的问题。

  

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  Calculating Combinations 
  ::计算组合

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  In other lessons, we have spent quite a bit of time learning how to count permutations , which are arrangements of items that take order into account.  Now we will be considering combinations , which are groups of items only concerned with which items are in the arrangement, regardless of the order.
  ::在其他教训中,我们花费了大量时间学习如何计算变数,这些变数是按顺序排列的项目安排。 现在我们将考虑组合,即只涉及安排中的项目的组合,不管顺序如何。

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  Naturally there are fewer unique combinations of a particular group of items than there are permutations, since there may be many permutations of the same single combination of items.  In fact, the formula for counting combinations without repetition is actually just a modification of the formula for permutations where the denominator is increased.  Take a look at the two formulas side-by-side:
  ::当然,某一组特定项目的独特组合比变式要少,因为同一单一组合项目可能有许多变异。 事实上,不重复计算组合的公式实际上只是对增加分母的变异公式的修改。 看看两个公式并列:

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  $$\begin{array}{r}Permutations:\frac{n!}{(n-r)!}Combinations:\frac{n!}{(n-r)!\times r!}\end{array}$$

  
  ::变异: n! ! 合并: n! !xr!

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  Where  $\begin{array}{r}n=\end{array}$ number of objects to choose from, and  $\begin{array}{r}r=\end{array}$ the number chosen
  ::n =要从中选择的对象数,r=所选择的数字

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  Notice how the combination formula is the same, with only the addition of the  $\begin{array}{r}r!\end{array}$ in the denominator of the fraction.  This makes sense conceptually, given that we need to reduce the number of arrangements that are the same items in a different order.
  ::注意组合公式是如何相同的, 只要在分数分母的分母中添加 r 。 这在概念上是有道理的, 因为我们需要按照不同的顺序减少相同项目的安排数量 。

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  In summary, to calculate the number of combinations possible from  $\begin{array}{r}n\end{array}$ items chosen $\begin{array}{r}r\end{array}$ at a time, use the combination formula: $\begin{array}{r}\frac{n!}{(n-r)!\times r!}\end{array}$
  ::简言之,为了计算一次从所选 n 项中选择 r 的组合数,请使用组合公式: n! (n-r)! xr!

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  Real-World Application: Ice Cream
  ::真实世界应用程序: 冰淇淋

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  The local ice cream shop carries twelve flavors.  How many different three-scoop bowls are possible?
  ::当地的冰淇淋店有12种口味 有多少不同的三勺碗?

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  There are twelve flavors, so $\begin{array}{r}n=12\end{array}$ .  We are creating bowls with three scoops each, so $\begin{array}{r}r=3\end{array}$ :
  ::有十二种口味, 所以N=12。我们各创造三勺的碗, 所以 r=3:

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  $$\begin{array}{rl}& \frac{12!}{(12-3)!\times 3!}\\ & \frac{12!}{9!\times 3!}\\ & \frac{12\times 11\times 10\times \bcancel{9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}}{\bcancel{(9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1)}(3\times 2\times 1)}(\text{cancel terms})\\ & \frac{1320}{6}\\ & 220differentflavors\end{array}$$

  
  ::12!!(12) x3) 3x3! 12!!9!x3! 12x11x11x10x9x8x7x6x6x5x4x3x3x2x1(9x8x7x6x5x4x3x3x2x1)(3x2x1) 13206220 不同口味

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  Real-World Application: Forming Groups
  ::现实世界应用:组成组

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  If there are thirteen students in a room, how many ways can a group of five be formed from them?
  ::如果一个房间里有13名学生,那么从他们那里可以形成多少种五人小组的方法?

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  There are thirteen students to choose from, so  $\begin{array}{r}n=13\end{array}$
  ::有13名学生可以选择,所以N=13

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  We are choosing five at a time, so  $\begin{array}{r}r=5\end{array}$
  ::我们每次选择五个, 所以 r=5

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  $$\begin{array}{rl}& \frac{13!}{(13-5)!\times 5!}\\ & \frac{13!}{(8!)(5!)}\\ & \frac{13\times 12\times 11\times 10\times 9}{5!}\\ & \frac{154440}{120}\\ & 1287\text{possible groups of five}\end{array}$$

  
  ::13! 13! 13! 13! 5! 5!) 13 12x11 10x10x95! 15444401201287 5个可能的组别

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  Real-World Application: Student Council
  ::现实世界应用:学生理事会

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  If your student council has seven members, how many ways are there to reach a majority vote?
  ::如果你的学生委员会有七名成员, 有多少方法可以达到多数票?

  

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  There are seven members to choose from, so $\begin{array}{r}n=7\end{array}$ .  A majority vote is either four, five, six, or seven votes, so we need to calculate the number of combinations for each $\begin{array}{r}r=4,5,6\end{array}$ , and 7, and add them together.
  ::7. 多数票为4票、5票、6票或7票,因此,我们需要计算每个r=4,5,6票和7票的组合数,并加在一起。

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  a. Four votes:
  ::a. 四票:

  $$\begin{array}{rl}& \frac{7!}{(7-4)!\times 4!}\\ & \frac{7\times 6\times 5\times 4\bcancel{\times 3\times 2\times 1}}{\bcancel{(3\times 2\times 1)}(4\times 3\times 2\times 1)}\\ & \frac{840}{24}=35\end{array}$$

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  b. Five votes:
  ::b. 五票:

  $$\begin{array}{rl}& \frac{7!}{(7-5)!\times (5!)}\\ & \frac{7\times 6\times 5\times 4\times 3}{5\times 4\times 3\times 2\times 1}\\ & \frac{2520}{120}=21\end{array}$$

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  c. Six votes:
  ::c. 六票:

  $$\begin{array}{rl}& \frac{7!}{(7-6)!\times 6!}\\ & \frac{7\times 6\times 5\times 4\times 3\times 2}{6\times 5\times 4\times 3\times 2\times 1}\\ & \frac{5040}{720}=7\end{array}$$

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  d. Seven votes:
  ::d. 七票:

  $$\begin{array}{rl}& \frac{7!}{(7-7)!\times 7!}\\ & \frac{7!}{7!}=1\end{array}$$

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  $\begin{array}{r}\text{Total}=35+21+7+1=64\end{array}$  ways to reach a majority vote
  ::共计=35+21+7+1=64 获得多数票的途径

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  Earlier  Problem Revisited
  ::重审先前的问题

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  Suppose there are fifteen students in a classroom, and you are looking to build a team of five students for a tug-of-war. How many different groups of five students could you create?
  ::假设一个教室里有15个学生,而你正在寻找建立一个由5个学生组成的团队来进行一场拔河战争。你能创造多少个由5个学生组成的不同群体呢?

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  This is a combinations problem requiring us to choose items five at a time from a pool of 15, which can be notated as 15C5 (read as "fifteen, choose five").  Use the combinations formula with n=15 and r=5:
  ::这是一个组合问题, 需要我们从 15 组中一次选择项目 5, 共 15 个, 共 15 个, 共 15 C 5 个( 读为“ 15 个, 选择 5 个 ” ) 。 使用 n= 15 和 r= 5 的组合公式 :

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  $$\begin{array}{rl}\text{Combinations}& =\frac{n!}{(n-r)!\times r!}\\ & =\frac{15!}{((15-5)!\times 5!)}\\ & =\frac{15!}{10!\times 5!}\\ & =\frac{15\times 14\times 13\times 12\times 11}{5!}\\ & =\frac{360,360}{120}\end{array}$$

  
  ::组合 ! (n-r) ! xr != 15! (( 15- 5) ! x5!) = 15. 10! x5 ! = 15x14x14x13x12x115! = 360, 360120

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  3,003 different teams
  ::3 003个不同小组

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  Examples 
  ::实例

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  Example 1
  ::例1

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  How many ways can six cars be chosen from a lot containing twenty-one cars?
  ::有多少方法可以选择六辆汽车? 从含有21辆汽车的很多汽车中挑选出多少方法?

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  There are twenty-one cars to choose from,  $\begin{array}{r}n=21\end{array}$
  ::有21辆汽车可供选择 N=21

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  There are six cars in each group, $\begin{array}{r}r=6\end{array}$
  ::每组有六辆车 r=6

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  $$\begin{array}{rl}& \frac{21!}{((21-6)!\times 6!)}\\ & \frac{21\times 20\times 19\times 18\times 17\times 16}{6!}\\ & 54264combinations\end{array}$$

  
  ::21!! (( 21- 6)! x6!) 21x20x19x18x18x17x166! 54264 组合

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  Example 2
  ::例2

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  How many unique groups of four can be made from nine pieces of candy?
  ::从九块糖果中可以制造出多少个独特的四组?

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  Nine pieces to choose from, $\begin{array}{r}n=9\end{array}$ , and four in each group, $\begin{array}{r}r=4\end{array}$ :
  ::从 n=9 中选取九块,每组选取四块,r=4:

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  $$\begin{array}{rl}& \frac{9!}{(9-4)!\times 4!}\\ & \frac{9\times 8\times 7\times 6}{4!}\\ & 756groups\end{array}$$

  
  ::9! (9- 4) !x4! 9x8x7x64. 756 组

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  Example 3
  ::例3

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  How many unique five-card hands are possible using a standard deck of cards?
  ::使用标准牌甲板有多少个独有的五卡手?

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  There are fifty-two card in a standard deck, $\begin{array}{r}n=52\end{array}$ , and five in each hand, $\begin{array}{r}r=5\end{array}$ :
  ::标准甲板有52张牌,N=52,每张牌有5张,r=5:

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  $$\begin{array}{rl}& \frac{52!}{(52-5)!\times 5!}\\ & \frac{52\times 51\times 50\times 49\times 48}{5!}\\ & 62,375,040possiblehands\end{array}$$

  
  ::52! (52-55)! x5! 52x51x50x49x49x485! 62375,040 可能的人

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  Review
  ::回顾

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  1. In the ball closet, there are 6 basketballs, footballs, and baseballs. You go in and pick 2 items at random. Assuming all the balls are marked differently, how many different groups of balls can you end up with?
     ::在球衣柜里,有6个篮球、足球和棒球。你随意进去取2个球项。假设所有的球的标记不同,你最后能选到多少种球?
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  2. You go to a store and buy 4 various items from 6 distinct choices. How many different possibilities were there?
     ::你去商店从6种不同的选择购买4种不同的物品。有多少不同的可能性?
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  3. Find the number of combinations of choosing 5 items from 9 distinct items.
     ::查找从9个不同项目中选择5个项目的组合数。
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  4. Find the number of combinations of choosing 6 items from 9 distinct items.
     ::从9个不同项目中查找选择 6 项的组合数 。
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  5. You have a pile of 7 distinctly different coins. How many ways can you take 2 coins from the pile?
     ::你有一堆7个截然不同的硬币 你能用多少种方式从硬币里拿2个硬币?
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  6. You are dealt a hand of 4 cards from a deck of 6 cards. Find the number of possible hands you can get
     ::从六张牌牌牌牌牌牌牌牌牌牌牌牌牌牌上用四张牌牌牌牌对您发牌。请查找您可以得到的手数
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  7. You hire 2 out of 9 people to build a garden. Find the number of possible ways of choosing your workers.
     ::你雇了9个人中的2个来建造花园 找出选择工人的可能方法
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  8. Find the number of combinations of choosing 4 items from 7 distinct items.
     ::从 7 个不同的项目中查找选择 4 个项目的组合数 。
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  9. Find the number of combinations of choosing 5 items from 5 distinct items.
     ::查找从5个不同项目中选择5个项目的组合数。
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  10. How many five-card hands are possible from a deck of twenty-six cards?
      ::26张牌牌的甲板上有多少五张牌手?
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  11. How many unique 7-person combinations are possible using 32 people?
      ::有多少单一的七人组合 可能使用32人?
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  12. If Pandi has nineteen chickens in her coop, how many unique groups of six chickens are possible from them?
      ::如果潘迪有十九只鸡 在她的房间, 有多少 独特的六只鸡群 可以从他们身上?
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  13. If one of her chickens lays colored eggs, every egg unique, how many four-color groups could she make from a dozen eggs?
      ::如果她的鸡之一产下彩蛋, 每个鸡蛋都独一无二, 她能用十几个鸡蛋做多少个四色组?

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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 。