求教一道排列组合数学题的思路

Helenmm · 发表于 2022-2-21 15:00:09

[size=13.333333015441895px]Six mobsters have arrived at the theater for the premiere of the film “Goodbuddies.” One of the mobsters, Frankie, is an informer, and he's afraid that another member of his crew, Joey, is on to him. Frankie, wanting to keep Joey in his sights, insists upon standing behind Joey in line at the concession stand, though not necessarily right behind him. How many ways can the six arrange themselves in line such that Frankie’s requirement is satisfied?
[size=13.333333015441895px] 6
[size=13.333333015441895px] 24
[size=13.333333015441895px] 120
[size=13.333333015441895px] 360
[size=13.333333015441895px] 720
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这题一开始想复杂了，因为说两个人不一定挨着，然后就开始分条件算所有的可能性，然后算到脑袋炸了，最后乱选一个。看完解析之后.....
Ignoring Frankie's requirement for a moment, observe that the six mobsters can be arranged 6! or 6 x 5 x 4 x 3 x 2 x 1 = 720 different ways in the concession stand line. In each of those 720 arrangements, Frankie must be either ahead of or behind Joey. Logically, since the combinations favor neither Frankie nor Joey, each would be behind the other in precisely half of the arrangements. Therefore, in order to satisfy Frankie's requirement, the six mobsters could be arranged in 720/2 = 360 different ways.

意思就是单独考虑两个人的睡前睡后的问题，各占一半，不用考虑其他几个。这种思路是怎样想到的呢？（这是一道600-700难度的题，郁闷
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敦敦酱 · 发表于 2022-2-21 15:36:06

遇到这种题可以先用ABC举例来找规律。
假如只有三个元素，我们需要b在a后面，有几种可能?
abc acb bac bca cab cba
仔细看，三个元素一共6种排列方式，任一一个元素在任一另一个元素后面有三种情况，比如
b在a后面三次，a在b后面三次。
大胆点，直接总结出规律是所有排列的一半，就是任一固定元素在另一个固定元素后面的排列数

Helenmm · 发表于 2022-2-21 18:13:29

敦敦酱发表于 2022-2-21 15:36
遇到这种题可以先用ABC举例来找规律。
假如只有三个元素，我们需要b在a后面，有几种可能?
abc acb bac bca ...

真机智！！好的！谢谢啊！

2022mbago · 发表于 2022-3-6 16:04:01

谢谢

[原始] 求教一道排列组合数学题的思路

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