定义和适用变数和因数

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Your job as county fair assistant is to arrange the sheep ribbons on the bulletin board overhanging the sheep pens. You have one Best in Show ribbon, one 1st Place ribbon, one 2nd Place ribbon, and one 3rd Place ribbon to display. How many ways can you arrange the ribbons?
::作为县集市助理,你的工作是安排布告板上的绵羊丝带,把羊圈挂在羊圈上。你有一个最佳的显示丝带,一个第一处显示带,一个第二处显示带,一个第三处显示带。你能用多少方法来安排丝带?

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Permutations and Factorials
::变差和因数

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The number of permutations of objects is the number of possible arrangements of the objects. Consider the following question: How many ways can seven DVD’s be arranged on a shelf? This is an example of a . We will use the without repetition to determine the permutations.
::对象的变换次数是对象的可能安排次数。 考虑下面的问题: 7个DVD在架子上可以排列多少种方式? 这是一个例子。 我们将使用不重复的方式来决定变换。

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Let's solve the following problem using permutations.
::让我们用变换来解决以下的问题。

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How many ways can 5 students sit in a row?
::5个学生能排成一排坐几条路?

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If we consider the students sitting in one of five seats, then we have 5 students to choose from for the first seat, four remaining to choose from for the second seat and so on until all the seats are filled.
::如果我们考虑到学生坐在五个席位中的一个,那么我们有五个学生从第一个席位中选择,四个学生从第二个席位中选择,等等,直到所有席位都得到填补。

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$\begin{align*}\underline5\times\underline4\times\underline3\times\underline2\times\underline1=120\end{align*}$ So there are 120 ways to seat the students.
::所以有120种办法让学生入校。

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The way we just wrote out $\begin{align*}\underline5\times\underline4\times\underline3\times\underline2\times\underline1\end{align*}$ can also be expressed as a . A factorial is the product of a number with each number less than itself. We use the notation, $\begin{align*}5!\end{align*}$ , which is read as “five factorial” to represent the expression $\begin{align*}\underline5\times\underline4\times\underline3\times\underline2\times\underline1\end{align*}$ . It is important to note that $\begin{align*}0!=1!=1\end{align*}$ . Students are often perplexed that both zero and one factorial are equal to one but think back to the context for illustration. If you want to arrange zero items, how many ways can you do it? If you want to arrange one item, how many ways can you do it? There is only one way to “arrange” zero or one item.
::我们刚刚写入 5 43321_ 的方式也可以表示为 。 阶乘是数字的产物, 每个数字比自己少。 我们使用符号 5! , 被读成“ 5 阶乘” 来表示 543321_ 。 需要注意的是 0! =1 =1. 学生们常常困惑地认为 0 和 1 阶乘 等同于 1, 但回想到上下文来说明 。 如果您想要安排零项, 您可以做多少方法 ? 如果您想要安排一个项目, 您可以做多少方法 ? 只有一个方法可以“ 排列 零 ” 或 1 项 。

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To evaluate a factorial on the TI-83 or 84 graphing calculator, type in the number, then press MATH $\begin{align*}\rightarrow\end{align*}$ NUM, $\begin{align*}4!\end{align*}$ . Press ENTER to evaluate.
::要对 TI-83 或 84 图形计算器进行阶乘评估, 请按 MATH NAUM, 4! 。 按 ENTER 来评估 。

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Now, let's solve the following problems using factorials.
::现在,让我们用乘数来解决以下的问题。

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1. Evaluate $\begin{align*}\frac{10!}{6!}\end{align*}$ .
   ::评估10.6!

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We should expand the numerator and the denominator to see which common factors the numerator and denominator that we can cancel out to simplify the expression.
::我们应扩大分子和分母,以了解哪些共同因素,即我们可以取消的分子和分母,以简化表达式。

$$\begin{align*}\frac{10\times9\times8\times7\times{\color{red}6\times5\times4\times3\times2\times1}}{{\color{red}6\times5\times4\times3\times2\times1}}=\frac{10\times9\times8\times7\times{\color{red}\cancel{6!}}}{{\color{red}\cancel{6!}}}=10\times9\times8\times7=5040\end{align*}$$

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2. On a shelf there are 6 different math books, 4 different science books and 8 novels. How many ways can the books be arranged if the groupings are maintained (meaning all the math books are together, the science books are together and the novels are together).
   ::书架上有6种不同的数学书籍、4种不同的科学书籍和8种小说。 如果组群维持不变,可以安排多少方法(这意味着所有数学书籍在一起,科学书籍在一起,小说在一起 ) 。

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There are 6 math books so if we think of filling 6 slots with the six books, then we start with 6 books for the first slot, then 5, then 4, etc: $\begin{align*}{\underline{{\color{red}6}}\times\underline{{\color{red}5}}\times\underline{{\color{red}4}}\times\underline{{\color{red}3}}\times\underline{{\color{red}2}}\times\underline{{\color{red}1}}}={\color{red}720}\end{align*}$ ways.
::有6本数学书 所以如果我们想用6本书填满6个空位, 那么我们从6本书开始, 开始第一个空位, 然后5本, 然后4本,等等: 654321720 种方式。

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There are 4 science books so we can arrange them in $\begin{align*}\underline{{\color{blue}4}}\times\underline{{\color{blue}3}}\times\underline{{\color{blue}2}}\times\underline{{\color{blue}1}}={\color{blue}24}\end{align*}$ ways.
::有4本科学书籍,所以我们可以用4+3+2+1+24的方式安排它们。

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There are 8 novels so we can arrange them in $\begin{align*}\underline{{\color{green}8}}\times\underline{{\color{green}7}}\times\underline{{\color{green}6}}\times\underline{{\color{green}5}}\times\underline{{\color{green}4}}\times\underline{{\color{green}3}}\times\underline{{\color{green}2}}\times\underline{{\color{green}1}}={\color{green}40,320}\end{align*}$ ways.
::有八本小说,所以我们可以用 87655432140,320的方式安排它们。

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Now, if each type of book can be arranged in so many ways and there are three types of books which can be displayed in $\begin{align*}\underline{3}\times\underline{2}\times\underline{1}=6\end{align*}$ ways, then there are:
::现在,如果每种书都能够以如此多的方式排列, 并且有三种书可以3216的方式展示, 那么有:

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$\begin{align*}{\color{red}720}\times{\color{blue}24}\times{\color{green}40320}\times6=4,180,377,600\end{align*}$ total ways to arrange the books.
::720x24x4040320x6=4,180,377,600 组织图书总方式。

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Examples
::实例

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Example 1
::例1

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Earlier, you were asked to find the number of  ways you can arrange the ribbons. 
::早些时候,有人要求你找到 多少方法可以安排丝带。

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If we consider the ribbons sitting in one of four spots, then we have four ribbons to choose from for the first spot, three remaining to choose from for the second spot and so on until all the spots are filled.
::如果我们考虑四点之一的丝带, 那么我们有四点的丝带 从第一点选择, 其余三点从第二点选择,等等,直到所有点都填满。

$\begin{align*}\underline4\times\underline3\times\underline2\times\underline1=24\end{align*}$

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Therefore, there are 24 ways to arrange the ribbons.
::因此,有24种方法可以安排丝带。

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For Examples 2 & 3, evaluate the expressions with factorials.
::例2和例3用系数来评价表达式。

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Example 2
::例2

$\begin{align*}\frac{12!}{9!}\end{align*}$

$\begin{align*}\frac{12\times11\times10\times{\color{red}\cancel{9!}}}{{\color{red}\cancel{9!}}}=1320\end{align*}$

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Example 3
::例3

$\begin{align*}\frac{4\times8!}{3!5!}\end{align*}$

$\begin{align*}\frac{4\times8\times7\times6\times{\color{red}\cancel{5!}}}{3\times2\times1\times{\color{red}\cancel{5!}}}=\frac{4\times8\times7\times{\color{red}\cancel{6}}}{{\color{red}\cancel{3}\times\cancel{2}}\times1}=224\end{align*}$

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Example 4
::例4

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How many ways can nine children line up?
::九个孩子能排成多少排队?

$\begin{align*}9!=362,880\end{align*}$

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Example 5
::例5

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How many ways can 3 cookbooks, 5 textbooks, 7 novels and 4 nonfiction books be arranged on a shelf if the groupings are maintained?
::如果保留分组,在架子上可以安排多少方法? 3本烹饪书籍、5本教科书、7本小说和4本非小说?

$\begin{align*}3!\times5!\times7!\times4!\times4!=2,090,188,800\end{align*}$

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Review
::回顾

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Evaluate the following factorial expressions.
::评估以下阶乘表达式。

1. $\begin{align*}\frac{5!}{2!3!}\end{align*}$
2. $\begin{align*}\frac{10!}{2!7!}\end{align*}$
3. $\begin{align*}\frac{4!8!}{9!}\end{align*}$
4. $\begin{align*}\frac{5!10!}{12!}\end{align*}$
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5. How many ways can a baseball team manager arrange nine players in a lineup?
   ::棒球队队长能用多少方法 安排九名球员排队?
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6. How many ways can the letters in the word FACTOR be arranged?
   ::能安排多少方法 字母的字词Fatortor?
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7. How many ways can 12 school buses line up?
   ::12辆校车能排几条路?
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8. How many ways can eight girls sit together in a row?
   ::8个女孩能坐在一起坐几条路?
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9. If the two of the eight girls in problem eight must sit together, how many ways can the 8 girls be arranged in the row such that the two girls sit together?
   ::如果问题8中的8个女孩中的2个必须坐在一起,那么8个女孩有多少方法可以排成一排,使两个女孩坐在一起?
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10. How many ways can seven diners sit around a circular table. (Hint: It is not $\begin{align*}7!\end{align*}$ , consider how a circular seating arrangement is different than a linear arrangement.)
    :提示:不是7!,请考虑循环座位安排与线性安排有何不同。 )
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11. How many ways can three cookbooks, four novels and two nonfiction books be arranged on a shelf if the groupings are maintained?
    ::如果保留分组,在一个架子上可以安排多少种方式? 三本烹饪书籍、四本小说和两本非小说。
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12. How many ways can two teachers, four male students, five female students and one administrator be arranged if the teachers must sit together, the male students must sit together and the female students must sit together?
    ::如果教师必须坐在一起,男学生必须坐在一起,女学生必须坐在一起,女学生必须坐在一起,那么可以安排如何安排两名教师、四名男学生、五名女学生和一名行政人员?

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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 。