Course: 8.7 使用技术 | The Way To Learn

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使用技术
Suppose you wanted to compare the number of possible permutations and combinations possible by choosing six, seven, or eight cards from either one or two standard card decks. Manually calculating all of those different cases could take quite a while. How could you use technology to simplify the process?
::假设您想通过从一个或两个标准纸牌牌牌上选择六、七或八张卡片来比较可能的变异和组合数量。人工计算所有这些不同情况需要很长一段时间。您如何使用技术简化程序?

Using Technology
::使用技术

Although calculating combinations and permutations ‘by hand’ is an excellent skill worthy of practice, since it is the best way to come to understand combinations and permutations, there are much more efficient ways to find the number of possibilities.
::虽然 " 亲手 " 计算组合和变相是一种极好的技巧,值得实践,因为它是理解组合和变相的最佳方法,但找到可能性数量的方法效率要高得多。

T he TI-84 calculator, which is very common in upper mathematics courses of all kinds, can the used to calculate combinatorics questions. Even more efficient, however, are some of the free and freely available online calculators. Here are a few examples:
::TI-84计算器在各种高级数学课程中都很常见,它可以用来计算组合解答问题。然而,效率更高的是一些免费和免费在线计算器。这里举几个例子:

"Math is Fun": Combinations and Permutations Calculator
::“母体是乐趣”:组合和交替计算器

This all-in-one combinatorics calculator is very simple to use, and can handle most basic combinatorics problems. The interface is self-explanatory, simply enter the number of items available to choose from (the $\begin{aligned} n \end{aligned}$ count), and the number to be chosen (the $\begin{aligned} r \end{aligned}$ count), specify if order is important ( or combination), and if repeats are allowed.
::此全在组合计算器非常容易使用, 并且可以处理最基本的组合计算器问题。界面不言自明, 只需输入可以从 n 中选择的项目数量, 以及要选择的项目数量( r 计数) , 具体说明顺序是否重要( 或组合) , 以及是否允许重复。

"Calculator Soup": Discrete Mathematics Calculator
::“计算器 Soup ” : 分辨数学计算器

Calculator Soup provides the ability to calculate more complex types of permutation and problems, but the interface is less user-friendly. Rather than providing an all-in-one calculator like ‘Math is Fun’, Calculator Soup has different calculators for each type of combination and permutation question. The screen shot below is from the basic combinations calculator. Note that the link above actually references the list of available combinatorics calculators, rather than any one specific calculator.
::计算器 Soup 提供了计算更复杂的变异类型和问题的能力, 但界面不那么方便用户。计算器 Soup 不是提供“ 数学就是乐趣” 这样的全在计算器, 而是为每种组合和变异问题提供不同的计算器。下面的屏幕截图来自基本组合计算器。请注意, 以上链接实际上引用了可用的组合计算器列表, 而不是任何一个特定的计算器。

"Joe Math": Factorials, Combinations, and Permutations Calculator
::“Joe Math”: 阶乘、组合和变数计算器

This last link references another basic combinations, , and permutations calculator. It does not appear to handle more complex problems, at least not on the surface, but it is offered here in case neither of the others is available.
::最后一个链接引用了另一个基本组合, 和调整计算器。它似乎没有处理更复杂的问题, 至少没有处理表面的问题, 但是这里提供它是为了以防其他两种组合都无法使用。

Evaluating Combinations
::评价组合

Use a calculator to evaluate $\begin{aligned} _{7} C_{4} \end{aligned}$ twice, once with repetitions allowed, and once without.
::使用计算器对 7C4 进行两次评估,一次允许重复,一次不重复。

Let’s use the Calculator soup calculator:
::让我们使用计算器汤计算器:

First, to calculate the number of combinations without repeats, choose the calculator labeled “ Combinations Calculator (nCr) ”. Enter “ 7 ” for $\begin{aligned} n \end{aligned}$ and “ 4 ” for $\begin{aligned} r \end{aligned}$ , and click “ calculate ”. Your screen should look like the image here.
::首先,为了计算不重复的组合数,请选择标有“复合计算器”(nCr)”的计算器。输入n和r的“7”和“4”,并单击“计算”。您的屏幕应该像图像一样。

There are 35 possible combinations without repeats.
::有35种可能的组合,没有重复。

Second, to calculate the number of combinations possible when repeats are allowed, choose the calculator labeled “ Combinations Replacement Calculator CR(n,r) ”. Again, enter “ 7 ” for “ $\begin{aligned} n \end{aligned}$ ”, and “ 4 ” for “ $\begin{aligned} r \end{aligned}$ ”, and click “ calculate ”. Your screen should look like the image here.
::其次,为了计算允许重复时可能的组合数,请选择标有“复合替换计算器CR(n,r)”的计算器。再次,输入“n”的“7”和“4”的“7”,单击“计算”。您屏幕应该像这里的图像。

There are 210 possible combinations with repeats.

::有210个可能的组合与重复。

Calculating the Number of Possible Permutations
::计算可能的变换数量

Use technology to calculate the number of possible permutations of the numbers 1-6 and letters A-F, both with and without repeats.
::使用技术计算数字1-6和字母A-F的可能排列次数,无论有无重复。

Let’s use the “Math is Fun” website this time:
::这次让我们使用“马思是乐趣”网站:

First, to calculate without repeats, enter “ 12 ” in both the “ Types to choose from ”, and “ Number chosen ” fields (since we want to know the number of permutations using all six numbers and all six letters), and set “ Is Order Important? ” to “ yes ”. Your screen should look like the image here.
::首先,不重复计算,在“从类型中选择”和“数字选择”字段中输入“12”(因为我们想知道使用所有六个数字和所有六个字母的变异次数),并将“顺序重要吗?”设为“是”。

There are apx 479,000,000 permutations possible without repeats.
::可能有4.79亿平方英尺的变异,无需重复。

Second, to calculate with repeats, all we need to do is change the dropdown menu labeled “ Is Repetition Allowed ” to “ yes ”. Your screen should show something similar to the image here.
::其次,用重复计算,我们需要做的就是将标签为“是否允许重复”的下调菜单改为“是 ” 。您的屏幕应该显示类似此图像的图像。

There are apx 8,916,100,000,000 permutations possible with repeats.
::有8 916 100 000 000阿皮克斯,重复时可能出现变异。

Evaluating Permutations
::评价差异

Use technology to evaluate $\begin{aligned} _{13} P_{11} \end{aligned}$ , both with and without repeats.
::使用技术对13P11进行评价,有重复和不重复。

Let’s use the “Calculator soup” calculator for this last one.
::让我们使用“计算汤”计算器来计算最后这一段。

To calculate without repeats, choose the “ Permutations Calculator (nPr) ” and enter “ 13 ” for $\begin{aligned} n \end{aligned}$ and “ 11 ” for $\begin{aligned} r \end{aligned}$ .
::要计算而不重复,请选择“调整计算器(nPr)”,输入n的“13”和r的“11”。

Click “ calculate ”, and you should see the output:
::点击“ 计算” , 您应该看到输出 :

$\begin{aligned} P (13, 11) = \frac{13!}{(13 - 11)!} = 3, 113, 510, 400 \end{aligned}$

::P(13,11)=13! (13-11)=3,113,510,400

To calculate with repeats, return to the list of statistics calculators, choose “ Permutations Replacement Calculator PR(n,r) ”, and enter “ 13 ” for $\begin{aligned} n \end{aligned}$ and “ 11 ” for $\begin{aligned} r \end{aligned}$ .
::要用重复计算,返回统计计算器列表,选择“变换替换计算器 PPR(n,r)”,并输入n的“13”和r的“11”。

Click “ calculate ”, and you should get:
::点击“计算”时,您应该得到:

$\begin{aligned} P R (13, 11) = n r = 1311 = 1, 792, 160, 394, 037 \end{aligned}$

::PR(13,11)=nr=1311=1,792,160,394,037

Earlier Problem Revisited
::重审先前的问题

Suppose you wanted to compare the number of possible permutations and combinations possible by choosing six, seven, or eight cards from either one or two standard card decks. Manually calculating all of those different cases could take quite a while. How could you use technology to simplify the process?
::假设您想通过从一个或两个标准纸牌牌牌上选择六、七或八张卡片来比较可能的变异和组合数量。人工计算所有这些不同情况需要很长一段时间。您如何使用技术简化程序?

By now, you should have no problems finding a technology resource to simplify this question. If you use the “ ” site, you just need to run through the various inputs for $\begin{aligned} n = 52 \end{aligned}$ (one deck) and $\begin{aligned} n = 104 \end{aligned}$ (two decks), $\begin{aligned} r = 6, 7 \end{aligned}$ , and 8 (for the number of cards chosen), and “ order matters ” = yes/no (to evaluate combinations and permutations).
::现在,您应该可以找到一个技术资源来简化这个问题。如果使用“ ” 网站,您只需要通过n=52 (一个甲板) 和 n=104 (两个甲板) 、 r= 6,7 和 8 (所选卡数) 以及“ 顺序事项” = 是/ 否 (用于评估组合和变换)。

Examples
::实例

Evaluate the combinations and permutations, you may use technology. We used the "Math is Fun" site.
::评估组合和变相, 你可以使用科技。我们使用“ 数学就是乐趣” 网站。

Example 1
::例1

$\begin{aligned} _{9} C_{5} \end{aligned}$ : No repeats
::9C5:无重复

$\begin{aligned} n = 9, r = 5 \end{aligned}$ , “Is Order Important” = no (since these are combinations), “Is Repetition Allowed” = no
::n=9,r=5,“顺序重要”=否(因为这些是组合),“是否允许重复”=否

There are 126 possible combinations
::有126种可能的组合

Example 2
::例2

$\begin{aligned} _{7} P_{6} \end{aligned}$ : Repeats allowed
::7P6: 允许重复

$\begin{aligned} n = 7, r = 6 \end{aligned}$ , “Is Order Important” = yes (since these are permutations), “Is Repetition Allowed” = yes
::n=7,r=6,“顺序重要”=是(因为这些是变式),“是否允许重复”=是

There are 117,649 possible permutations
::可能有117,649个可能的变换

Example 3
::例3

$\begin{aligned} _{10} C_{2} \end{aligned}$ : Repeats allowed
::10C2:允许重复

$\begin{aligned} n = 10, r = 2 \end{aligned}$ , “Is Order Important” = no (combinations), “Is Repetition Allowed” = yes
::n=10,r=2,“顺序重要”=否(组合),“是否允许再次”=是

There are 55 unique combinations
::有55个独特的组合

Example 4
::例4

$\begin{aligned} _{8} P_{4} \end{aligned}$ : No repeats
::8P4:无重复

$\begin{aligned} n = 8, r = 4 \end{aligned}$ , “Is Order Important” = yes (permutations), “Is Repetition Allowed” = no
::n=8,r=4,“顺序重要”=是(调整),“是否允许重复”=否

There are 1680 unique permutations
::有1 680个独特的变式

Review
::回顾

Evaluate the combinations and permutations, you may use technology.
::评估组合和组合,您可以使用技术。

$\begin{aligned} _{5} C_{3} \end{aligned}$ : No repeats
::5C3:无重复

$\begin{aligned} _{12} P_{6} \end{aligned}$ : Repeats allowed
::12P6:允许重复

$\begin{aligned} _{8} C_{3} \end{aligned}$ : Repeats allowed
::8C3:允许重复

$\begin{aligned} _{18} P_{17} \end{aligned}$ : No repeats
::18P17:无重复

$\begin{aligned} _{9} C_{9} \end{aligned}$ : No repeats
::9C9:无重复

$\begin{aligned} _{7} P_{7} \end{aligned}$ : Repeats allowed
::7P7:允许重复

$\begin{aligned} _{10} C_{10} \end{aligned}$ : Repeats allowed
::10C10:允许重复

$\begin{aligned} _{3} P_{12} \end{aligned}$ : Repeats allowed
::3P12:允许重复

$\begin{aligned} _{5} C_{9} \end{aligned}$ : Repeats allowed
::5C9:允许重复

$\begin{aligned} _{4} P_{6} \end{aligned}$ : Repeats allowed
::4P6:允许重复

$\begin{aligned} _{3} C_{21} \end{aligned}$ : Repeats allowed
::3C21:允许重复

$\begin{aligned} _{6} P_{3} \end{aligned}$ : No repeats
::6P3:无重复

$\begin{aligned} _{6} C_{15} \end{aligned}$ : Repeats allowed
::6C15:允许重复

$\begin{aligned} _{7} P_{61} \end{aligned}$ : Repeats allowed
::7P61:允许重复

$\begin{aligned} _{10} C_{2} \end{aligned}$ : No repeats
::10C2:无重复

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。

8.7 使用技术

Section outline

General

使用技术

Using Technology
::使用技术

"Math is Fun": Combinations and Permutations Calculator
::“母体是乐趣”:组合和交替计算器

"Calculator Soup": Discrete Mathematics Calculator
::“计算器 Soup ” : 分辨数学计算器

"Joe Math": Factorials, Combinations, and Permutations Calculator
::“Joe Math”: 阶乘、组合和变数计算器

Evaluating Combinations
::评价组合

Calculating the Number of Possible Permutations
::计算可能的变换数量

Evaluating Permutations
::评价差异

Earlier Problem Revisited
::重审先前的问题

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)

Section outline

General

使用技术

Using Technology ::使用技术

"Math is Fun": Combinations and Permutations Calculator ::“母体是乐趣”:组合和交替计算器

"Calculator Soup": Discrete Mathematics Calculator ::“计算器 Soup ” : 分辨数学计算器

"Joe Math": Factorials, Combinations, and Permutations Calculator ::“Joe Math”: 阶乘、组合和变数计算器

Evaluating Combinations ::评价组合

Calculating the Number of Possible Permutations ::计算可能的变换数量

Evaluating Permutations ::评价差异

Earlier Problem Revisited ::重审先前的问题

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Using Technology
::使用技术

"Math is Fun": Combinations and Permutations Calculator
::“母体是乐趣”:组合和交替计算器

"Calculator Soup": Discrete Mathematics Calculator
::“计算器 Soup ” : 分辨数学计算器

"Joe Math": Factorials, Combinations, and Permutations Calculator
::“Joe Math”: 阶乘、组合和变数计算器

Evaluating Combinations
::评价组合

Calculating the Number of Possible Permutations
::计算可能的变换数量

Evaluating Permutations
::评价差异

Earlier Problem Revisited
::重审先前的问题

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)