【Java/算法/排列】某单位安排7名员工在十一期间值班，每人值班一天。若7名员工中的甲乙排在相邻两天，丙不排在10月1日，丁不排在10月7日。请列举出所有安排方案？

【原题干】

某单位安排7名员工在十一期间值班，每人值班一天。若7名员工中的甲乙排在相邻两天，丙不排在10月1日，丁不排在10月7日。问安排方案共有几种？（此题为10年重庆市高考理科第9题）

【数学解法】

首先总的无限制全排列方式有A77=5040种；

甲乙既然相邻便作为一个整体考虑，其内部有甲乙、乙甲两种方式；

甲乙捆绑与丙丁戊己庚排列有A66*2种，这是捆绑甲乙后的无限制全排总数；

将丙安排在十月一日，其余甲乙捆绑与丁戊己庚全排，有A55*2种；

将丁安排在十月七日，其余甲乙捆绑与丙戊己庚全排，也有A55*2种；

将丙安排在十月一日，将丁安排在十月七日，其余甲乙捆绑与戊己庚全排，有A44*2种；这是减去后要补回来的。

最后，结果=A66*2-2*A55*2+A44*2=1008种

【代码解法】

思路：将甲乙丙丁戊己庚全排，然后按三个条件过滤就可以了。

代码：

package test230421;

import java.util.ArrayList;
import java.util.List;

public class Arrange4 {
    static List<List<Integer>> results = new ArrayList<>();
    
    public static void main(String[] args) {
        // 七人名称
        final String[] names= {"甲","乙","丙","丁","戊","己","庚"};
        
        // 七人对应的脚标
        int[] arr = {0,1,2,3,4,5,6};
  
        Arrange4 h1 = new Arrange4();
        // 获得全排列链表
        h1.dfs(arr,new ArrayList<>());
        
        int idx=0;
        for (List<Integer> ls : results) {
            // 1.过滤条件：丙不在十月一日
            if(ls.get(0)==2) {
                // 第一个脚标是丙对应的脚标2
                continue;
            }
            
            // 2.过滤条件：丁不在十月七日
            if(ls.get(6)==3) {
                // 第一个脚标是丁对应的脚标3
                continue;
            }
            
            // 由脚标链表得到名称链表
            List<String> joinedNames=new ArrayList<String>(names.length);
            for(int i=0;i<names.length;i++) {
                joinedNames.add(names[ls.get(i)]);
            }
            String joined=String.join(",", joinedNames);
            
            // 3.过滤条件：甲乙必相邻
            if(joined.contains("甲,乙") || joined.contains("乙,甲")) {
                System.out.println(String.format("%04d",++idx)+"."+joined);
            }
            
        }  
    }
  
    public List<List<Integer>> dfs( int[] arr,List<Integer> list){
        List<Integer> temp = new ArrayList<>(list);
        if (arr.length == list.size()){
            results.add(temp);
        }
        for (int i=0;i<arr.length;i++){
            if (temp.contains(arr[i])){
                continue;
            }
            temp.add(arr[i]);
            dfs(arr,temp);
            temp.remove(temp.size()-1);
        }
        return results;
    }
}

【结果】：

0001.甲,乙,丙,丁,戊,己,庚
0002.甲,乙,丙,丁,戊,庚,己
0003.甲,乙,丙,丁,己,戊,庚
0004.甲,乙,丙,丁,己,庚,戊
0005.甲,乙,丙,丁,庚,戊,己
0006.甲,乙,丙,丁,庚,己,戊
0007.甲,乙,丙,戊,丁,己,庚
0008.甲,乙,丙,戊,丁,庚,己
0009.甲,乙,丙,戊,己,丁,庚
0010.甲,乙,丙,戊,庚,丁,己
0011.甲,乙,丙,己,丁,戊,庚
0012.甲,乙,丙,己,丁,庚,戊
0013.甲,乙,丙,己,戊,丁,庚
0014.甲,乙,丙,己,庚,丁,戊
0015.甲,乙,丙,庚,丁,戊,己
0016.甲,乙,丙,庚,丁,己,戊
0017.甲,乙,丙,庚,戊,丁,己
0018.甲,乙,丙,庚,己,丁,戊
0019.甲,乙,丁,丙,戊,己,庚
0020.甲,乙,丁,丙,戊,庚,己
0021.甲,乙,丁,丙,己,戊,庚
0022.甲,乙,丁,丙,己,庚,戊
0023.甲,乙,丁,丙,庚,戊,己
0024.甲,乙,丁,丙,庚,己,戊
0025.甲,乙,丁,戊,丙,己,庚
0026.甲,乙,丁,戊,丙,庚,己
0027.甲,乙,丁,戊,己,丙,庚
0028.甲,乙,丁,戊,己,庚,丙
0029.甲,乙,丁,戊,庚,丙,己
0030.甲,乙,丁,戊,庚,己,丙
0031.甲,乙,丁,己,丙,戊,庚
0032.甲,乙,丁,己,丙,庚,戊
0033.甲,乙,丁,己,戊,丙,庚
0034.甲,乙,丁,己,戊,庚,丙
0035.甲,乙,丁,己,庚,丙,戊
0036.甲,乙,丁,己,庚,戊,丙
0037.甲,乙,丁,庚,丙,戊,己
0038.甲,乙,丁,庚,丙,己,戊
0039.甲,乙,丁,庚,戊,丙,己
0040.甲,乙,丁,庚,戊,己,丙
0041.甲,乙,丁,庚,己,丙,戊
0042.甲,乙,丁,庚,己,戊,丙
0043.甲,乙,戊,丙,丁,己,庚
0044.甲,乙,戊,丙,丁,庚,己
0045.甲,乙,戊,丙,己,丁,庚
0046.甲,乙,戊,丙,庚,丁,己
0047.甲,乙,戊,丁,丙,己,庚
0048.甲,乙,戊,丁,丙,庚,己
0049.甲,乙,戊,丁,己,丙,庚
0050.甲,乙,戊,丁,己,庚,丙
0051.甲,乙,戊,丁,庚,丙,己
0052.甲,乙,戊,丁,庚,己,丙
0053.甲,乙,戊,己,丙,丁,庚
0054.甲,乙,戊,己,丁,丙,庚
0055.甲,乙,戊,己,丁,庚,丙
0056.甲,乙,戊,己,庚,丁,丙
0057.甲,乙,戊,庚,丙,丁,己
0058.甲,乙,戊,庚,丁,丙,己
0059.甲,乙,戊,庚,丁,己,丙
0060.甲,乙,戊,庚,己,丁,丙
0061.甲,乙,己,丙,丁,戊,庚
0062.甲,乙,己,丙,丁,庚,戊
0063.甲,乙,己,丙,戊,丁,庚
0064.甲,乙,己,丙,庚,丁,戊
0065.甲,乙,己,丁,丙,戊,庚
0066.甲,乙,己,丁,丙,庚,戊
0067.甲,乙,己,丁,戊,丙,庚
0068.甲,乙,己,丁,戊,庚,丙
0069.甲,乙,己,丁,庚,丙,戊
0070.甲,乙,己,丁,庚,戊,丙
0071.甲,乙,己,戊,丙,丁,庚
0072.甲,乙,己,戊,丁,丙,庚
0073.甲,乙,己,戊,丁,庚,丙
0074.甲,乙,己,戊,庚,丁,丙
0075.甲,乙,己,庚,丙,丁,戊
0076.甲,乙,己,庚,丁,丙,戊
0077.甲,乙,己,庚,丁,戊,丙
0078.甲,乙,己,庚,戊,丁,丙
0079.甲,乙,庚,丙,丁,戊,己
0080.甲,乙,庚,丙,丁,己,戊
0081.甲,乙,庚,丙,戊,丁,己
0082.甲,乙,庚,丙,己,丁,戊
0083.甲,乙,庚,丁,丙,戊,己
0084.甲,乙,庚,丁,丙,己,戊
0085.甲,乙,庚,丁,戊,丙,己
0086.甲,乙,庚,丁,戊,己,丙
0087.甲,乙,庚,丁,己,丙,戊
0088.甲,乙,庚,丁,己,戊,丙
0089.甲,乙,庚,戊,丙,丁,己
0090.甲,乙,庚,戊,丁,丙,己
0091.甲,乙,庚,戊,丁,己,丙
0092.甲,乙,庚,戊,己,丁,丙
0093.甲,乙,庚,己,丙,丁,戊
0094.甲,乙,庚,己,丁,丙,戊
0095.甲,乙,庚,己,丁,戊,丙
0096.甲,乙,庚,己,戊,丁,丙
0097.乙,甲,丙,丁,戊,己,庚
0098.乙,甲,丙,丁,戊,庚,己
0099.乙,甲,丙,丁,己,戊,庚
0100.乙,甲,丙,丁,己,庚,戊
0101.乙,甲,丙,丁,庚,戊,己
0102.乙,甲,丙,丁,庚,己,戊
0103.乙,甲,丙,戊,丁,己,庚
0104.乙,甲,丙,戊,丁,庚,己
0105.乙,甲,丙,戊,己,丁,庚
0106.乙,甲,丙,戊,庚,丁,己
0107.乙,甲,丙,己,丁,戊,庚
0108.乙,甲,丙,己,丁,庚,戊
0109.乙,甲,丙,己,戊,丁,庚
0110.乙,甲,丙,己,庚,丁,戊
0111.乙,甲,丙,庚,丁,戊,己
0112.乙,甲,丙,庚,丁,己,戊
0113.乙,甲,丙,庚,戊,丁,己
0114.乙,甲,丙,庚,己,丁,戊
0115.乙,甲,丁,丙,戊,己,庚
0116.乙,甲,丁,丙,戊,庚,己
0117.乙,甲,丁,丙,己,戊,庚
0118.乙,甲,丁,丙,己,庚,戊
0119.乙,甲,丁,丙,庚,戊,己
0120.乙,甲,丁,丙,庚,己,戊
0121.乙,甲,丁,戊,丙,己,庚
0122.乙,甲,丁,戊,丙,庚,己
0123.乙,甲,丁,戊,己,丙,庚
0124.乙,甲,丁,戊,己,庚,丙
0125.乙,甲,丁,戊,庚,丙,己
0126.乙,甲,丁,戊,庚,己,丙
0127.乙,甲,丁,己,丙,戊,庚
0128.乙,甲,丁,己,丙,庚,戊
0129.乙,甲,丁,己,戊,丙,庚
0130.乙,甲,丁,己,戊,庚,丙
0131.乙,甲,丁,己,庚,丙,戊
0132.乙,甲,丁,己,庚,戊,丙
0133.乙,甲,丁,庚,丙,戊,己
0134.乙,甲,丁,庚,丙,己,戊
0135.乙,甲,丁,庚,戊,丙,己
0136.乙,甲,丁,庚,戊,己,丙
0137.乙,甲,丁,庚,己,丙,戊
0138.乙,甲,丁,庚,己,戊,丙
0139.乙,甲,戊,丙,丁,己,庚
0140.乙,甲,戊,丙,丁,庚,己
0141.乙,甲,戊,丙,己,丁,庚
0142.乙,甲,戊,丙,庚,丁,己
0143.乙,甲,戊,丁,丙,己,庚
0144.乙,甲,戊,丁,丙,庚,己
0145.乙,甲,戊,丁,己,丙,庚
0146.乙,甲,戊,丁,己,庚,丙
0147.乙,甲,戊,丁,庚,丙,己
0148.乙,甲,戊,丁,庚,己,丙
0149.乙,甲,戊,己,丙,丁,庚
0150.乙,甲,戊,己,丁,丙,庚
0151.乙,甲,戊,己,丁,庚,丙
0152.乙,甲,戊,己,庚,丁,丙
0153.乙,甲,戊,庚,丙,丁,己
0154.乙,甲,戊,庚,丁,丙,己
0155.乙,甲,戊,庚,丁,己,丙
0156.乙,甲,戊,庚,己,丁,丙
0157.乙,甲,己,丙,丁,戊,庚
0158.乙,甲,己,丙,丁,庚,戊
0159.乙,甲,己,丙,戊,丁,庚
0160.乙,甲,己,丙,庚,丁,戊
0161.乙,甲,己,丁,丙,戊,庚
0162.乙,甲,己,丁,丙,庚,戊
0163.乙,甲,己,丁,戊,丙,庚
0164.乙,甲,己,丁,戊,庚,丙
0165.乙,甲,己,丁,庚,丙,戊
0166.乙,甲,己,丁,庚,戊,丙
0167.乙,甲,己,戊,丙,丁,庚
0168.乙,甲,己,戊,丁,丙,庚
0169.乙,甲,己,戊,丁,庚,丙
0170.乙,甲,己,戊,庚,丁,丙
0171.乙,甲,己,庚,丙,丁,戊
0172.乙,甲,己,庚,丁,丙,戊
0173.乙,甲,己,庚,丁,戊,丙
0174.乙,甲,己,庚,戊,丁,丙
0175.乙,甲,庚,丙,丁,戊,己
0176.乙,甲,庚,丙,丁,己,戊
0177.乙,甲,庚,丙,戊,丁,己
0178.乙,甲,庚,丙,己,丁,戊
0179.乙,甲,庚,丁,丙,戊,己
0180.乙,甲,庚,丁,丙,己,戊
0181.乙,甲,庚,丁,戊,丙,己
0182.乙,甲,庚,丁,戊,己,丙
0183.乙,甲,庚,丁,己,丙,戊
0184.乙,甲,庚,丁,己,戊,丙
0185.乙,甲,庚,戊,丙,丁,己
0186.乙,甲,庚,戊,丁,丙,己
0187.乙,甲,庚,戊,丁,己,丙
0188.乙,甲,庚,戊,己,丁,丙
0189.乙,甲,庚,己,丙,丁,戊
0190.乙,甲,庚,己,丁,丙,戊
0191.乙,甲,庚,己,丁,戊,丙
0192.乙,甲,庚,己,戊,丁,丙
0193.丁,甲,乙,丙,戊,己,庚
0194.丁,甲,乙,丙,戊,庚,己
0195.丁,甲,乙,丙,己,戊,庚
0196.丁,甲,乙,丙,己,庚,戊
0197.丁,甲,乙,丙,庚,戊,己
0198.丁,甲,乙,丙,庚,己,戊
0199.丁,甲,乙,戊,丙,己,庚
0200.丁,甲,乙,戊,丙,庚,己
0201.丁,甲,乙,戊,己,丙,庚
0202.丁,甲,乙,戊,己,庚,丙
0203.丁,甲,乙,戊,庚,丙,己
0204.丁,甲,乙,戊,庚,己,丙
0205.丁,甲,乙,己,丙,戊,庚
0206.丁,甲,乙,己,丙,庚,戊
0207.丁,甲,乙,己,戊,丙,庚
0208.丁,甲,乙,己,戊,庚,丙
0209.丁,甲,乙,己,庚,丙,戊
0210.丁,甲,乙,己,庚,戊,丙
0211.丁,甲,乙,庚,丙,戊,己
0212.丁,甲,乙,庚,丙,己,戊
0213.丁,甲,乙,庚,戊,丙,己
0214.丁,甲,乙,庚,戊,己,丙
0215.丁,甲,乙,庚,己,丙,戊
0216.丁,甲,乙,庚,己,戊,丙
0217.丁,乙,甲,丙,戊,己,庚
0218.丁,乙,甲,丙,戊,庚,己
0219.丁,乙,甲,丙,己,戊,庚
0220.丁,乙,甲,丙,己,庚,戊
0221.丁,乙,甲,丙,庚,戊,己
0222.丁,乙,甲,丙,庚,己,戊
0223.丁,乙,甲,戊,丙,己,庚
0224.丁,乙,甲,戊,丙,庚,己
0225.丁,乙,甲,戊,己,丙,庚
0226.丁,乙,甲,戊,己,庚,丙
0227.丁,乙,甲,戊,庚,丙,己
0228.丁,乙,甲,戊,庚,己,丙
0229.丁,乙,甲,己,丙,戊,庚
0230.丁,乙,甲,己,丙,庚,戊
0231.丁,乙,甲,己,戊,丙,庚
0232.丁,乙,甲,己,戊,庚,丙
0233.丁,乙,甲,己,庚,丙,戊
0234.丁,乙,甲,己,庚,戊,丙
0235.丁,乙,甲,庚,丙,戊,己
0236.丁,乙,甲,庚,丙,己,戊
0237.丁,乙,甲,庚,戊,丙,己
0238.丁,乙,甲,庚,戊,己,丙
0239.丁,乙,甲,庚,己,丙,戊
0240.丁,乙,甲,庚,己,戊,丙
0241.丁,丙,甲,乙,戊,己,庚
0242.丁,丙,甲,乙,戊,庚,己
0243.丁,丙,甲,乙,己,戊,庚
0244.丁,丙,甲,乙,己,庚,戊
0245.丁,丙,甲,乙,庚,戊,己
0246.丁,丙,甲,乙,庚,己,戊
0247.丁,丙,乙,甲,戊,己,庚
0248.丁,丙,乙,甲,戊,庚,己
0249.丁,丙,乙,甲,己,戊,庚
0250.丁,丙,乙,甲,己,庚,戊
0251.丁,丙,乙,甲,庚,戊,己
0252.丁,丙,乙,甲,庚,己,戊
0253.丁,丙,戊,甲,乙,己,庚
0254.丁,丙,戊,甲,乙,庚,己
0255.丁,丙,戊,乙,甲,己,庚
0256.丁,丙,戊,乙,甲,庚,己
0257.丁,丙,戊,己,甲,乙,庚
0258.丁,丙,戊,己,乙,甲,庚
0259.丁,丙,戊,己,庚,甲,乙
0260.丁,丙,戊,己,庚,乙,甲
0261.丁,丙,戊,庚,甲,乙,己
0262.丁,丙,戊,庚,乙,甲,己
0263.丁,丙,戊,庚,己,甲,乙
0264.丁,丙,戊,庚,己,乙,甲
0265.丁,丙,己,甲,乙,戊,庚
0266.丁,丙,己,甲,乙,庚,戊
0267.丁,丙,己,乙,甲,戊,庚
0268.丁,丙,己,乙,甲,庚,戊
0269.丁,丙,己,戊,甲,乙,庚
0270.丁,丙,己,戊,乙,甲,庚
0271.丁,丙,己,戊,庚,甲,乙
0272.丁,丙,己,戊,庚,乙,甲
0273.丁,丙,己,庚,甲,乙,戊
0274.丁,丙,己,庚,乙,甲,戊
0275.丁,丙,己,庚,戊,甲,乙
0276.丁,丙,己,庚,戊,乙,甲
0277.丁,丙,庚,甲,乙,戊,己
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0926.庚,丁,己,甲,乙,戊,丙
0927.庚,丁,己,乙,甲,丙,戊
0928.庚,丁,己,乙,甲,戊,丙
0929.庚,丁,己,丙,甲,乙,戊
0930.庚,丁,己,丙,乙,甲,戊
0931.庚,丁,己,丙,戊,甲,乙
0932.庚,丁,己,丙,戊,乙,甲
0933.庚,丁,己,戊,甲,乙,丙
0934.庚,丁,己,戊,乙,甲,丙
0935.庚,丁,己,戊,丙,甲,乙
0936.庚,丁,己,戊,丙,乙,甲
0937.庚,戊,甲,乙,丙,丁,己
0938.庚,戊,甲,乙,丁,丙,己
0939.庚,戊,甲,乙,丁,己,丙
0940.庚,戊,甲,乙,己,丁,丙
0941.庚,戊,乙,甲,丙,丁,己
0942.庚,戊,乙,甲,丁,丙,己
0943.庚,戊,乙,甲,丁,己,丙
0944.庚,戊,乙,甲,己,丁,丙
0945.庚,戊,丙,甲,乙,丁,己
0946.庚,戊,丙,乙,甲,丁,己
0947.庚,戊,丙,丁,甲,乙,己
0948.庚,戊,丙,丁,乙,甲,己
0949.庚,戊,丙,丁,己,甲,乙
0950.庚,戊,丙,丁,己,乙,甲
0951.庚,戊,丙,己,丁,甲,乙
0952.庚,戊,丙,己,丁,乙,甲
0953.庚,戊,丁,甲,乙,丙,己
0954.庚,戊,丁,甲,乙,己,丙
0955.庚,戊,丁,乙,甲,丙,己
0956.庚,戊,丁,乙,甲,己,丙
0957.庚,戊,丁,丙,甲,乙,己
0958.庚,戊,丁,丙,乙,甲,己
0959.庚,戊,丁,丙,己,甲,乙
0960.庚,戊,丁,丙,己,乙,甲
0961.庚,戊,丁,己,甲,乙,丙
0962.庚,戊,丁,己,乙,甲,丙
0963.庚,戊,丁,己,丙,甲,乙
0964.庚,戊,丁,己,丙,乙,甲
0965.庚,戊,己,甲,乙,丁,丙
0966.庚,戊,己,乙,甲,丁,丙
0967.庚,戊,己,丙,丁,甲,乙
0968.庚,戊,己,丙,丁,乙,甲
0969.庚,戊,己,丁,甲,乙,丙
0970.庚,戊,己,丁,乙,甲,丙
0971.庚,戊,己,丁,丙,甲,乙
0972.庚,戊,己,丁,丙,乙,甲
0973.庚,己,甲,乙,丙,丁,戊
0974.庚,己,甲,乙,丁,丙,戊
0975.庚,己,甲,乙,丁,戊,丙
0976.庚,己,甲,乙,戊,丁,丙
0977.庚,己,乙,甲,丙,丁,戊
0978.庚,己,乙,甲,丁,丙,戊
0979.庚,己,乙,甲,丁,戊,丙
0980.庚,己,乙,甲,戊,丁,丙
0981.庚,己,丙,甲,乙,丁,戊
0982.庚,己,丙,乙,甲,丁,戊
0983.庚,己,丙,丁,甲,乙,戊
0984.庚,己,丙,丁,乙,甲,戊
0985.庚,己,丙,丁,戊,甲,乙
0986.庚,己,丙,丁,戊,乙,甲
0987.庚,己,丙,戊,丁,甲,乙
0988.庚,己,丙,戊,丁,乙,甲
0989.庚,己,丁,甲,乙,丙,戊
0990.庚,己,丁,甲,乙,戊,丙
0991.庚,己,丁,乙,甲,丙,戊
0992.庚,己,丁,乙,甲,戊,丙
0993.庚,己,丁,丙,甲,乙,戊
0994.庚,己,丁,丙,乙,甲,戊
0995.庚,己,丁,丙,戊,甲,乙
0996.庚,己,丁,丙,戊,乙,甲
0997.庚,己,丁,戊,甲,乙,丙
0998.庚,己,丁,戊,乙,甲,丙
0999.庚,己,丁,戊,丙,甲,乙
1000.庚,己,丁,戊,丙,乙,甲
1001.庚,己,戊,甲,乙,丁,丙
1002.庚,己,戊,乙,甲,丁,丙
1003.庚,己,戊,丙,丁,甲,乙
1004.庚,己,戊,丙,丁,乙,甲
1005.庚,己,戊,丁,甲,乙,丙
1006.庚,己,戊,丁,乙,甲,丙
1007.庚,己,戊,丁,丙,甲,乙
1008.庚,己,戊,丁,丙,乙,甲

【参考资料】

1.《新编中学数学解题方法1000招丛书【排列和组合】》P98 刘培杰工作室编 哈尔滨工业大学出版社出版

2.https://www.jb51.net/article/252294.htm

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