这个问题推广一下,就是错排问题,是组合数学中的问题之一。考虑一个有n个元素的排列,若一个排列中所有的元素都不在自己原来的位置上,那么这样的排列就称为原排列的一个错排。 n个元素的错排数记为D(n)。 研究一个排列错排个数的问题,叫做错排问题或称为更列问题
OK,现在详细分析这个问题,我们要的最终结果就是书的编号与所在位置的编号不同,在这里,我们把n本书的错排操作数记为D(n),那n-1本就是D(n-1),n-2本就是D(n-2)啦,下面,我们把放置问题分为两步(初始位置号与书的编号相同):
第一步:
我们取一本书,书的编号为m,现在这本书就在我们手中,注意,按照题目要求,最开始的时候这本书的位置号也是m号,按照题目要求,我们现在放书时不能放回这个位置m了,而是要选择其他位置,那么有多少种选择呢,想一下,总共有n本书,n个位置,现在我手里这本书不能把它放到位置m,那么剩下的n-1个位置我当然就是随便扔啦,也就是n-1种扔法,好,现在,我选择了位置k,我决定把手里这本书放到位置k这里,记住这个是位置编号k,那么,我肯定要把原来这里的编号为k的书拿出来,再把这本编号为n的书放进去喽。所以,现在我们手里的书的编号是k
第二步:
我们把手里这本编号为k的书本放到书架,注意,放的过程中我们又面临两种情况,可以想到,此时此刻现在书架上编号m的位置是空着的,所以我们可以选择放在这个位置上,书的编号为k,位置编号为m,没错,满足题意,这是第一种情况,还有一种就是我不选择这个空着的位置m,我再重新选择一个新的位置,我们称之为第二种情况,下面详细分析
第一种情况:我把这本编号为k的书放到这个编号为m的地址,那现在我们面前是什么状况呢,就是位置k和位置m的书交换位置,也就是位置号不等于书号,即满足错排,总共n个位置,我们只动了m和k这两个位置,那么剩下的n-2个位置还是纹丝不动,保持一一对应的关系呢,那么对于剩下的这n-2本书的错排操作,我们又回到了问题的起点,求n-2本的错排操作数D(n-2),结合第一步,我们可以得到第一种情况总共有(n-1)*D(n-2)种方法
第二种情况:我们不选择这个空着的位置m啦,我们手持这本编号为k的书,我们从除了位置m以及位置k的剩下的n-2个位置中选择一个位置,OK,我们现在开始想,我手里这本书不能放在这个位置m,嗯嗯,除了第一步我们放置的那本书m不用管了,我们还要搞手里这本和剩下的n-2本,也就是n-1本,同时又要求手里这本k还不能放到位置m,这是不是就相当于把手里这本加上剩下的n-2本也就是n-1本书进行错排呢,哇哇哇,想一想,错排的定义,要求每本书都不能呆在某一个特定位置,是不是刚好符合呢qwq,所以,现在的为题就到了求手里这本和剩下的n-2本总共是n-1本书的错排操作数,我们记为D(n-1),结合第一步,我们得出这第二种情况共有(n-1)*D(n-1)种方法
好的,现在我们总结两种情况,结果进行相加,就可以得到递推公式啦
递推公式为:
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题目不是要求每一本书都和原来的书的位置相同,现在不仅仅满足2本书的位置不相同而已嘛举报回复
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版权声明:本文为博主原创文章,未经博主允许不得转载。 所谓的错排,有一个错排公式可以来运用,但是就个人而言,还是想给大家深入讲解其含义,当然:“如果看到此处觉得没有必要再深入去了解,自己只是需要个... 博文 来自: Lucky_-的博客
I-没人中奖概率--错排公式
09-08 阅读数 490
HDU 2006'10 ACM contest的颁奖晚会隆重开始了! 为了活跃气氛,组织者举行了一个别开生面、奖品丰厚的抽奖活动,这个活动的具体要求是这样的: 首先,所有参加晚会的人员都将一张写有自己... 博文 来自: JingleLiA的博客
错排公式浅谈(推导+应用)
12-12 阅读数 13
给出一个已经排好的长度为n的数列,问全部排错一共有多少排法?典型利用错排公式去解决问题,(搜一下)我们不难知道错排公式递推式为
在中国程序员是青春饭吗?
03-17 阅读数 15万+
今年,我也32了 ,为了不给大家误导,咨询了猎头、圈内好友,以及年过35岁的几位老程序员……舍了老脸去揭人家伤疤……希望能给大家以帮助,记得帮我点赞哦。目录:你以为的人生 一次又一次的伤害 猎头界的真... 博文 来自: 启舰
大学四年,我决定把Java学习过的书籍都分享一遍
03-25 阅读数 1万+
给岁月以文明,而不是给文明以岁月,技术人读书我觉得很有必要,那这份书单的大部分书我觉得对您有用。... 博文 来自: 敖丙
卸载 x 雷某度!GitHub 标星 1.5w+,从此我只用这款全能高速下载工具!
12-29 阅读数 16万+
作者 | Rocky0429来源 | Python空间大家好,我是 Rocky0429,一个喜欢在网上收集各种资源的蒟蒻…网上资源眼花缭乱,下载的方式也同样千奇百怪,比如 BT 下载,磁力链接,网... 博文 来自: Rocky0429
为什么猝死的都是程序员,基本上不见产品经理猝死呢?
01-05 阅读数 11万+
相信大家时不时听到程序员猝死的消息,但是基本上听不到产品经理猝死的消息,这是为什么呢?我们先百度搜一下:程序员猝死,出现将近700多万条搜索结果:搜索一下:产品经理猝死,只有400万条的搜索结果,从搜... 博文 来自: 曹银飞的专栏
推荐10个堪称神器的学习网站
02-25 阅读数 25万+
每天都会收到很多读者的私信,问我:“二哥,有什么推荐的学习网站吗?最近很浮躁,手头的一些网站都看烦了,想看看二哥这里有什么新鲜货。”今天一早做了个恶梦,梦到被老板辞退了。虽然说在我们公司,只有我辞退老... 博文 来自: 沉默王二
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