如果总共有 25 名学生，老师可以用多少种方式安排前排的 10 名学生？

在数学中，排列是指组织集合的行为，其中集合的所有成员都排列成某种顺序或顺序。换句话说，如果集合已经被排列，那么它的组件的重新排列称为置换过程。几乎所有数学领域都以或多或少的重要方式发生排列。当考虑某些有限集上的不同命令时，它们经常出现。

置换公式

排列是从一组 n 个事物中选择 r 个事物而无需替换，并且顺序很重要。

ⁿ P _r = (n!)/(nr)!

什么是组合？

组合是从组中选择项目的行为，这样（与排列不同）选择顺序无关紧要。在较小的情况下，可以计算合并的数量。组合是指一次取 k 的 n 样东西不重复地合并。为了指代允许重复出现的组合，经常使用术语 k-selection 或 k-combination with repeat。

组合配方

组合是从 n 个事物的集合中选择 r 个事物而无需替换且顺序无关紧要。

$_{r}^{n}\textrm{C} = \binom{n}{r} =\ \frac{_{r}^{n}\textrm{P}}{r!} = \frac{n!}{r!(n-r)!}$

如果总共有 25 名学生，老师可以用多少种方式安排前排的 10 名学生？

解决方案：-

Here in this problem, we have to determine the total number of ways in which we can

arrange some number of students out of certain students in the front row of the

classroom. First of all, we will determine the total number of ways in which we can

select 10 students out of 25 number of students. Then we will permute or arrange them.

No. of ways of selecting r things out of n is = (n¦r)

The number of ways in which we can arrange r number of different things = r!

Answer and Explanation:

Total number of students = 25

Number of places in the front row = 10

Number of ways in which 10 students can be arranged in a row is 10!. This is because

each student can be seated in any of the 10 seats. There could be 3628800 possible

arrangements.

Number of ways selecting 10 students out of 25 = (25¦10)

类似问题

问题 1：有 8 名男性和 10 名女性，您需要组成一个由 5 名男性和 6 名女性组成的委员会。
委员会可以通过几种方式组成？

回答：-

We need to select 5 men from 8 men and 6 women from 10 women.

Number of ways to do this

= ⁸C₅ × ¹⁰C₆

= ⁸C₃ × ¹⁰C₄ [∵ ⁿC_r = ⁿC_(n-r)]

= [(8 × 7 × 6)/(3 × 2 × 1)] × [(10 × 9 × 8 × 7)/(4 × 3 × 2 × 1)]

= 56 × 210

= 11760

问题 2：共有 10 人组成一个 5 人的团队，每个团队应该包括两个特定的人，有多少种方法可以组成一个团队？

回答：

Two particular persons should be included in each team. Therefore we have to select the

remaining (5 – 2) = 3 persons from (10 – 2) = 8 persons.

Hence, the required number of ways

= ⁸C₃  

= (8×7×6)/(3×2×1)

= 8 × 7

= 56