8男8女排成一排，性别交替，有多少种方式？
在数学中，排列被称为排列一个集合的过程，其中一个集合的所有成员被排列成一些系列或顺序。如果集合已经排列，则排列的过程称为对其组件的重新排列。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。
什么是组合？
组合是从组中选择项目的行为，这样（不像排列）选择的顺序无关紧要。在较小的情况下，可以计算组合的数量。组合是指一次取k个不重复的n个事物的并集。组合可以以任意顺序选择项目。对于那些允许重复出现的组合，经常使用术语 k-selection 或 k-combination with replication。
置换公式
在排列中，从 n 个事物的集合中选择 r 个事物，没有任何替换。在这个选择的顺序。
nPr = (n!) / (n-r)!
Here,
n = set size, the total number of items in the set
r = subset size , the number of items to be selected from the set
组合配方
组合 r 个事物是从一组 n 个事物中选择的，其中选择的顺序无关紧要。
nCr = n!/(n−r)!r!
Here,
n = Number of items in set
r = Number of items selected from the set
8男8女排成一排，性别交替，有多少种方式？
解决方案：
8 men’s, 8 women’s
1) even places by women, odd places by men
There are 8 number of ways for men standing in the first position, similarly there are
8 no ways for women standing in second position then there are 7 number of ways for men standing in the 3rd position and there are 7 number of ways for women standing in the 4th position and so on…until all the places are filled.
So, there are 8 number of ways for men standing in the odd position and for women standing in the even position
Number of ways
⇒ 8! × 8!
⇒ (8!)²
2) even places by men, odd places by women
There are 8 number of ways for women standing in the first position, similarly there are 8 no ways for men standing in second position then there are 7 number of ways for women standing in the 3rd position and there are 7 number of ways for men standing in the 4th position and so on…until all the places are filled.
So, there are 8 number of ways for men standing in the even position and for women standing in the odd position
Number of ways
⇒ 8! × 8!
⇒ (8!)²
Total number of ways
⇒ (8!)² + (8!)²
⇒ 2 × (8!)²
类似问题
问题一：7男7女有几种方式可以站成一排，但是没有女的可以站在一起？
解决方案：
7 men’s, 7 women’s
1) even places by women, odd places by men
There are 7 number of ways for men standing in the first position, similarly there are 7 no ways for women standing in second position then there are 6 number of ways for men standing in the 3rd position and there are 6 number of ways for women standing in the 4th position and so on…until all the places are filled.
So, there are 7 number of ways for men standing in the odd position and 7 no of ways for women standing in the even position
Number of ways
⇒ 7! × 7!
⇒ (7!)²
2) even places by men, odd places by women
There are 7 number of ways for women standing in the first position, similarly there are 7 no ways for men standing in second position then there are 6 number of ways for women standing in the 3rd position and there are 6 number of ways for men standing in the 4th position and so on…until all the places are filled.
So, there are 7 number of ways for women standing in the odd position and 7 no of ways for men standing in the even position
Number of ways
⇒ 7! × 7!
⇒ (7!)²
Total number of ways
⇒ (7!)² + (7!)²
⇒ 2 × (7!)²