在大小为n的给定数组中打印r元素的所有可能组合

给定大小为n的数组，生成并打印数组中r个元素的所有可能组合。例如，如果输入数组为{1、2、3、4}，且r为2，则输出应为{1、2}，{1、3}，{1、4}，{2、3}，{ 2，4}和{3，4}。
以下是执行此操作的两种方法。
方法1(修复元素并重复出现)
我们创建了一个临时数组’data []’，它逐一存储所有输出。这个想法是从data []中的第一个索引(index = 0)开始，在该索引处一个接一个的固定元素，然后对剩余的索引进行递归。令输入数组为{1、2、3、4、5}，r为3。我们首先将1固定在data []中的索引0处，然后递归其余索引，然后将2固定在索引0处并递归。最后，我们修复3并递归剩余的索引。当data []中的元素数等于r(组合的大小)时，我们将打印data []。
下图显示了相同输入的递归树。

以下是上述方法的实施。

C++

// C++ program to print all combination
// of size r in an array of size n
#include
using namespace std;
 
void combinationUtil(int arr[], int data[],
                    int start, int end,
                    int index, int r);
 
// The main function that prints
// all combinations of size r
// in arr[] of size n. This function
// mainly uses combinationUtil()
void printCombination(int arr[], int n, int r)
{
    // A temporary array to store
    // all combination one by one
    int data[r];
 
    // Print all combination using
    // temprary array 'data[]'
    combinationUtil(arr, data, 0, n-1, 0, r);
}
 
/* arr[] ---> Input Array
data[] ---> Temporary array to
store current combination
start & end ---> Staring and
Ending indexes in arr[]
index ---> Current index in data[]
r ---> Size of a combination to be printed */
void combinationUtil(int arr[], int data[],
                    int start, int end,
                    int index, int r)
{
    // Current combination is ready
    // to be printed, print it
    if (index == r)
    {
        for (int j = 0; j < r; j++)
            cout << data[j] << " ";
        cout << endl;
        return;
    }
 
    // replace index with all possible
    // elements. The condition "end-i+1 >= r-index"
    // makes sure that including one element
    // at index will make a combination with
    // remaining elements at remaining positions
    for (int i = start; i <= end &&
        end - i + 1 >= r - index; i++)
    {
        data[index] = arr[i];
        combinationUtil(arr, data, i+1,
                        end, index+1, r);
    }
}
 
// Driver code
int main()
{
    int arr[] = {1, 2, 3, 4, 5};
    int r = 3;
    int n = sizeof(arr)/sizeof(arr[0]);
    printCombination(arr, n, r);
}
 
// This code is contributed by rathbhupendra

C

// Program to print all combination of size r in an array of size n
#include 
void combinationUtil(int arr[], int data[], int start, int end,
                     int index, int r);
 
// The main function that prints all combinations of size r
// in arr[] of size n. This function mainly uses combinationUtil()
void printCombination(int arr[], int n, int r)
{
    // A temporary array to store all combination one by one
    int data[r];
 
    // Print all combination using temprary array 'data[]'
    combinationUtil(arr, data, 0, n-1, 0, r);
}
 
/* arr[]  ---> Input Array
   data[] ---> Temporary array to store current combination
   start & end ---> Staring and Ending indexes in arr[]
   index  ---> Current index in data[]
   r ---> Size of a combination to be printed */
void combinationUtil(int arr[], int data[], int start, int end,
                     int index, int r)
{
    // Current combination is ready to be printed, print it
    if (index == r)
    {
        for (int j=0; j= r-index" makes sure that including one element
    // at index will make a combination with remaining elements
    // at remaining positions
    for (int i=start; i<=end && end-i+1 >= r-index; i++)
    {
        data[index] = arr[i];
        combinationUtil(arr, data, i+1, end, index+1, r);
    }
}
 
// Driver program to test above functions
int main()
{
    int arr[] = {1, 2, 3, 4, 5};
    int r = 3;
    int n = sizeof(arr)/sizeof(arr[0]);
    printCombination(arr, n, r);
}

Java

// Java program to print all combination of size r in an array of size n
import java.io.*;
 
class Combination {
 
    /* arr[]  ---> Input Array
    data[] ---> Temporary array to store current combination
    start & end ---> Staring and Ending indexes in arr[]
    index  ---> Current index in data[]
    r ---> Size of a combination to be printed */
    static void combinationUtil(int arr[], int data[], int start,
                                int end, int index, int r)
    {
        // Current combination is ready to be printed, print it
        if (index == r)
        {
            for (int j=0; j= r-index" makes sure that including one element
        // at index will make a combination with remaining elements
        // at remaining positions
        for (int i=start; i<=end && end-i+1 >= r-index; i++)
        {
            data[index] = arr[i];
            combinationUtil(arr, data, i+1, end, index+1, r);
        }
    }
 
    // The main function that prints all combinations of size r
    // in arr[] of size n. This function mainly uses combinationUtil()
    static void printCombination(int arr[], int n, int r)
    {
        // A temporary array to store all combination one by one
        int data[]=new int[r];
 
        // Print all combination using temprary array 'data[]'
        combinationUtil(arr, data, 0, n-1, 0, r);
    }
 
    /*Driver function to check for above function*/
    public static void main (String[] args) {
        int arr[] = {1, 2, 3, 4, 5};
        int r = 3;
        int n = arr.length;
        printCombination(arr, n, r);
    }
}
 
/* This code is contributed by Devesh Agrawal */

Python3

# Program to print all combination
# of size r in an array of size n
 
# The main function that prints
# all combinations of size r in
# arr[] of size n. This function
# mainly uses combinationUtil()
def printCombination(arr, n, r):
     
    # A temporary array to
    # store all combination
    # one by one
    data = [0]*r;
 
    # Print all combination
    # using temprary array 'data[]'
    combinationUtil(arr, data, 0,
                    n - 1, 0, r);
 
# arr[] ---> Input Array
# data[] ---> Temporary array to
#         store current combination
# start & end ---> Staring and Ending
#             indexes in arr[]
# index ---> Current index in data[]
# r ---> Size of a combination
# to be printed
def combinationUtil(arr, data, start,
                    end, index, r):
                         
    # Current combination is ready
    # to be printed, print it
    if (index == r):
        for j in range(r):
            print(data[j], end = " ");
        print();
        return;
 
    # replace index with all
    # possible elements. The
    # condition "end-i+1 >=
    # r-index" makes sure that
    # including one element at
    # index will make a combination
    # with remaining elements at
    # remaining positions
    i = start;
    while(i <= end and end - i + 1 >= r - index):
        data[index] = arr[i];
        combinationUtil(arr, data, i + 1,
                        end, index + 1, r);
        i += 1;
 
# Driver Code
arr = [1, 2, 3, 4, 5];
r = 3;
n = len(arr);
printCombination(arr, n, r);
 
# This code is contributed by mits

C#

// C# program to print all
// combination of size r
// in an array of size n
using System;
 
class GFG
{
    /* arr[] ---> Input Array
    data[] ---> Temporary array to
                store current combination
    start & end ---> Staring and Ending
                     indexes in arr[]
    index ---> Current index in data[]
    r ---> Size of a combination
            to be printed */
    static void combinationUtil(int []arr, int []data,
                                int start, int end,
                                int index, int r)
    {
        // Current combination is
        // ready to be printed,
        // print it
        if (index == r)
        {
            for (int j = 0; j < r; j++)
                Console.Write(data[j] + " ");
            Console.WriteLine("");
            return;
        }
 
        // replace index with all
        // possible elements. The
        // condition "end-i+1 >=
        // r-index" makes sure that
        // including one element
        // at index will make a
        // combination with remaining
        // elements at remaining positions
        for (int i = start; i <= end &&
                  end - i + 1 >= r - index; i++)
        {
            data[index] = arr[i];
            combinationUtil(arr, data, i + 1,
                            end, index + 1, r);
        }
    }
 
    // The main function that prints
    // all combinations of size r
    // in arr[] of size n. This
    // function mainly uses combinationUtil()
    static void printCombination(int []arr,
                                 int n, int r)
    {
        // A temporary array to store
        // all combination one by one
        int []data = new int[r];
 
        // Print all combination
        // using temprary array 'data[]'
        combinationUtil(arr, data, 0,
                        n - 1, 0, r);
    }
 
    // Driver Code
    static public void Main ()
    {
        int []arr = {1, 2, 3, 4, 5};
        int r = 3;
        int n = arr.Length;
        printCombination(arr, n, r);
    }
}
 
// This code is contributed by m_kit

PHP

 Input Array
data[] ---> Temporary array to
            store current combination
start & end ---> Staring and Ending
                 indexes in arr[]
index ---> Current index in data[]
r ---> Size of a combination
       to be printed */
function combinationUtil($arr, $data, $start,
                         $end, $index, $r)
                 
{
    // Current combination is ready
    // to be printed, print it
    if ($index == $r)
    {
        for ($j = 0; $j < $r; $j++)
            echo $data[$j];
        echo "\n";
        return;
    }
 
    // replace index with all
    // possible elements. The
    // condition "end-i+1 >=
    // r-index" makes sure that
    // including one element at
    // index will make a combination
    // with remaining elements at
    // remaining positions
    for ($i = $start;
         $i <= $end &&
         $end - $i + 1 >= $r - $index; $i++)
    {
        $data[$index] = $arr[$i];
        combinationUtil($arr, $data, $i + 1,
                        $end, $index + 1, $r);
    }
}
 
// Driver Code
$arr = array(1, 2, 3, 4, 5);
$r = 3;
$n = sizeof($arr);
printCombination($arr, $n, $r);
 
// This code is contributed by ajit
?>

Javascript

 
// Javascript program to print all
// combination of size r in an array of size n  
 
    /* arr[]  ---> Input Array
    data[] ---> Temporary array to store current combination
    start & end ---> Staring and Ending indexes in arr[]
    index  ---> Current index in data[]
    r ---> Size of a combination to be printed */
    function combinationUtil(arr,data,start,end,index,r)
    {
        // Current combination is ready to be printed, print it
        if (index == r)
        {
            for (let j=0; j<r; j++)
            {
                document.write(data[j]+" ");
            }
            document.write("<br>")
        }
         
        // replace index with all possible elements. The condition
        // "end-i+1 >= r-index" makes sure that including one element
        // at index will make a combination with remaining elements
        // at remaining positions
        for (let i=start; i<=end && end-i+1 >= r-index; i++)
        {
            data[index] = arr[i];
            combinationUtil(arr, data, i+1, end, index+1, r);
        }
    }
     
    // The main function that prints all combinations of size r
    // in arr[] of size n. This function mainly uses combinationUtil()
    function printCombination(arr,n,r)
    {
        // A temporary array to store all combination one by one
        let data = new Array(r);
         
        // Print all combination using temprary array 'data[]'
        combinationUtil(arr, data, 0, n-1, 0, r);
    }
     
    /*Driver function to check for above function*/
    let arr=[1, 2, 3, 4, 5];
    let r = 3;
    let n = arr.length;
    printCombination(arr, n, r);
     
     
    // This code is contributed by rag2127
     

输出：

1 2 3
1 2 4
1 2 5
1 3 4
1 3 5
1 4 5
2 3 4
2 3 5
2 4 5
3 4 5

如何处理重复项?
请注意，上述方法不处理重复项。例如，如果输入数组为{1、2、1}，而r为2，则程序将{1、2}和{2、1}打印为两个不同的组合。我们可以通过在上面的代码中添加以下两点来避免重复。
1)在printCombination()中调用combinationUtil()之前，添加代码以对数组进行排序
2)在combinationUtil()的for循环末尾添加以下行

// Since the elements are sorted, all occurrences of an element
        // must be together
        while (arr[i] == arr[i+1])
             i++; 

该手柄重复实现看到这一点。
方法2(包括和排除每个元素)
与上述方法一样，我们创建一个临时数组data []。这里的想法类似于子集和问题。我们一个接一个地考虑输入数组的每个元素，然后重复两种情况：
1)元素包含在当前组合中(我们将元素放入data []中，并在data []中增加下一个可用索引)
2)当前组合中不包含该元素(我们不放置该元素并且不更改索引)
当data []中的元素数等于r(组合的大小)时，我们将其打印出来。
该方法主要基于Pascal的Identity，即n _cr = n-1 _cr + n-1 _cr-1
以下是方法2的实现。

C++

// C++ Program to print all combination of
// size r in an array of size n
#include 
using namespace std;
void combinationUtil(int arr[], int n, int r,
                    int index, int data[], int i);
 
// The main function that prints all
// combinations of size r in arr[]
// of size n. This function mainly
// uses combinationUtil()
void printCombination(int arr[], int n, int r)
{
    // A temporary array to store
    // all combination one by one
    int data[r];
 
    // Print all combination using
    // temprary array 'data[]'
    combinationUtil(arr, n, r, 0, data, 0);
}
 
/* arr[] ---> Input Array
n ---> Size of input array
r ---> Size of a combination to be printed
index ---> Current index in data[]
data[] ---> Temporary array to store current combination
i ---> index of current element in arr[] */
void combinationUtil(int arr[], int n, int r,
                    int index, int data[], int i)
{
    // Current combination is ready, print it
    if (index == r)
    {
        for (int j = 0; j < r; j++)
            cout << data[j] << " ";
        cout << endl;
        return;
    }
 
    // When no more elements are there to put in data[]
    if (i >= n)
        return;
 
    // current is included, put next at next location
    data[index] = arr[i];
    combinationUtil(arr, n, r, index + 1, data, i + 1);
 
    // current is excluded, replace it with next (Note that
    // i+1 is passed, but index is not changed)
    combinationUtil(arr, n, r, index, data, i+1);
}
 
// Driver code
int main()
{
    int arr[] = {1, 2, 3, 4, 5};
    int r = 3;
    int n = sizeof(arr)/sizeof(arr[0]);
    printCombination(arr, n, r);
    return 0;
}
 
// This is code is contributed by rathbhupendra

C

// Program to print all combination of size r in an array of size n
#include
void combinationUtil(int arr[],int n,int r,int index,int data[],int i);
 
// The main function that prints all combinations of size r
// in arr[] of size n. This function mainly uses combinationUtil()
void printCombination(int arr[], int n, int r)
{
    // A temporary array to store all combination one by one
    int data[r];
 
    // Print all combination using temprary array 'data[]'
    combinationUtil(arr, n, r, 0, data, 0);
}
 
/* arr[]  ---> Input Array
   n      ---> Size of input array
   r      ---> Size of a combination to be printed
   index  ---> Current index in data[]
   data[] ---> Temporary array to store current combination
   i      ---> index of current element in arr[]     */
void combinationUtil(int arr[], int n, int r, int index, int data[], int i)
{
    // Current cobination is ready, print it
    if (index == r)
    {
        for (int j=0; j= n)
        return;
 
    // current is included, put next at next location
    data[index] = arr[i];
    combinationUtil(arr, n, r, index+1, data, i+1);
 
    // current is excluded, replace it with next (Note that
    // i+1 is passed, but index is not changed)
    combinationUtil(arr, n, r, index, data, i+1);
}
 
// Driver program to test above functions
int main()
{
    int arr[] = {1, 2, 3, 4, 5};
    int r = 3;
    int n = sizeof(arr)/sizeof(arr[0]);
    printCombination(arr, n, r);
    return 0;
}

Java

// Java program to print all combination of size r in an array of size n
import java.io.*;
 
class Combination {
 
    /* arr[]  ---> Input Array
    data[] ---> Temporary array to store current combination
    start & end ---> Staring and Ending indexes in arr[]
    index  ---> Current index in data[]
    r ---> Size of a combination to be printed */
    static void combinationUtil(int arr[], int n, int r, int index,
                                int data[], int i)
    {
        // Current combination is ready to be printed, print it
        if (index == r)
        {
            for (int j=0; j= n)
        return;
 
        // current is included, put next at next location
        data[index] = arr[i];
        combinationUtil(arr, n, r, index+1, data, i+1);
 
        // current is excluded, replace it with next (Note that
        // i+1 is passed, but index is not changed)
        combinationUtil(arr, n, r, index, data, i+1);
    }
 
    // The main function that prints all combinations of size r
    // in arr[] of size n. This function mainly uses combinationUtil()
    static void printCombination(int arr[], int n, int r)
    {
        // A temporary array to store all combination one by one
        int data[]=new int[r];
 
        // Print all combination using temprary array 'data[]'
        combinationUtil(arr, n, r, 0, data, 0);
    }
 
    /*Driver function to check for above function*/
    public static void main (String[] args) {
        int arr[] = {1, 2, 3, 4, 5};
        int r = 3;
        int n = arr.length;
        printCombination(arr, n, r);
    }
}
/* This code is contributed by Devesh Agrawal */

的Python 3

# Program to print all combination
# of size r in an array of size n
 
# The main function that prints all
# combinations of size r in arr[] of
# size n. This function mainly uses
# combinationUtil()
def printCombination(arr, n, r):
 
    # A temporary array to store
    # all combination one by one
    data = [0] * r
 
    # Print all combination using
    # temprary array 'data[]'
    combinationUtil(arr, n, r, 0, data, 0)
 
''' arr[] ---> Input Array
n     ---> Size of input array
r     ---> Size of a combination to be printed
index ---> Current index in data[]
data[] ---> Temporary array to store
            current combination
i     ---> index of current element in arr[]     '''
def combinationUtil(arr, n, r, index, data, i):
 
    # Current cobination is ready,
    # print it
    if (index == r):
        for j in range(r):
            print(data[j], end = " ")
        print()
        return
 
    # When no more elements are
    # there to put in data[]
    if (i >= n):
        return
 
    # current is included, put
    # next at next location
    data[index] = arr[i]
    combinationUtil(arr, n, r, index + 1,
                    data, i + 1)
 
    # current is excluded, replace it
    # with next (Note that i+1 is passed,
    # but index is not changed)
    combinationUtil(arr, n, r, index,
                    data, i + 1)
 
# Driver Code
if __name__ == "__main__":
    arr = [1, 2, 3, 4, 5]
    r = 3
    n = len(arr)
    printCombination(arr, n, r)
 
# This code is contributed
# by ChitraNayal

C＃

// C# program to print all
// combination of size r
// in an array of size n
using System;
 
class GFG
{
     
    /* arr[] ---> Input Array
    data[] ---> Temporary array to
                store current combination
    start & end ---> Staring and Ending
                     indexes in arr[]
    index ---> Current index in data[]
    r ---> Size of a combination
           to be printed */
    static void combinationUtil(int []arr, int n,
                                int r, int index,
                                int []data, int i)
    {
        // Current combination is ready
        // to be printed, print it
        if (index == r)
        {
            for (int j = 0; j < r; j++)
                Console.Write(data[j] + " ");
                Console.WriteLine("");
            return;
        }
 
        // When no more elements are
        // there to put in data[]
        if (i >= n)
        return;
 
        // current is included, put
        // next at next location
        data[index] = arr[i];
        combinationUtil(arr, n, r,
                        index + 1, data, i + 1);
 
        // current is excluded, replace
        // it with next (Note that
        // i+1 is passed, but index
        // is not changed)
        combinationUtil(arr, n, r, index,
                        data, i + 1);
    }
 
    // The main function that prints
    // all combinations of size r
    // in arr[] of size n. This
    // function mainly uses combinationUtil()
    static void printCombination(int []arr,
                                 int n, int r)
    {
        // A temporary array to store
        // all combination one by one
        int []data = new int[r];
 
        // Print all combination
        // using temprary array 'data[]'
        combinationUtil(arr, n, r, 0, data, 0);
    }
 
    // Driver Code
    static public void Main ()
    {
        int []arr = {1, 2, 3, 4, 5};
        int r = 3;
        int n = arr.Length;
        printCombination(arr, n, r);
    }
}
 
// This code is contributed by ajit

的PHP

 Input Array
n ---> Size of input array
r ---> Size of a combination
       to be printed
index ---> Current index in data[]
data[] ---> Temporary array to store
            current combination
i ---> index of current element in arr[] */
function combinationUtil($arr, $n, $r,
                         $index, $data, $i)
{
    // Current cobination
    // is ready, print it
    if ($index == $r)
    {
        for ($j = 0; $j < $r; $j++)
            echo $data[$j], " ";
        echo "\n";
        return;
    }
 
    // When no more elements are
    // there to put in data[]
    if ($i >= $n)
        return;
 
    // current is included, put
    // next at next location
    $data[$index] = $arr[$i];
    combinationUtil($arr, $n, $r,
                    $index + 1,
                    $data, $i + 1);
 
    // current is excluded, replace
    // it with next (Note that i+1
    // is passed, but index is not changed)
    combinationUtil($arr, $n, $r,
                    $index, $data, $i + 1);
}
 
// Driver Code
$arr = array(1, 2, 3, 4, 5);
$r = 3;
$n = sizeof($arr);
printCombination($arr, $n, $r);
 
// This code is contributed by ajit
?>

Java脚本

 
// Javascript program to print all
// combination of size r in an array of size n   
     
    /* arr[]  ---> Input Array
    data[] ---> Temporary array to
    store current combination
    start & end ---> Staring and
    Ending indexes in arr[]
    index  ---> Current index in data[]
    r ---> Size of a combination to be printed */
    function combinationUtil(arr,n,r,index,data,i)
    {
        // Current combination is ready
        // to be printed, print it
        if (index == r)
        {
            for (let j=0; j<r; j++)
            {
                document.write(data[j]+" ");
            }
            document.write("<br>");
         
            return;
        }
         
        // When no more elements are there
        // to put in data[]
        if (i >= n)
        {
            return;
        }
         
        // current is included, put
        // next at next location
        data[index] = arr[i];
        combinationUtil(arr, n, r, index+1, data, i+1);
  
        // current is excluded, replace
        // it with next (Note that
        // i+1 is passed, but index is not changed)
        combinationUtil(arr, n, r, index, data, i+1);
         
    }
     
    // The main function that prints
    // all combinations of size r
    // in arr[] of size n. This function
    // mainly uses combinationUtil()
    function printCombination(arr,n,r)
    {
        // A temporary array to store
        // all combination one by one
        let data=new Array(r);
         
        // Print all combination using
        // temprary array 'data[]'
        combinationUtil(arr, n, r, 0, data, 0);
    }
     
    /*Driver function to check for above function*/
    let arr=[1, 2, 3, 4, 5];
    let r = 3;
    let n = arr.length;
    printCombination(arr, n, r);
     
    // This code is contributed by avanitrachhadiya2155
     

输出 ：

1 2 3
1 2 4
1 2 5
1 3 4
1 3 5
1 4 5
2 3 4
2 3 5
2 4 5
3 4 5

如何处理方法2中的重复项?
像方法1一样，我们可以遵循两件事来处理重复项。
1)在printCombination()中调用combinationUtil()之前，添加代码以对数组进行排序
2)在combinationUtil()的combinationUtil()的两个递归调用之间添加以下行

// Since the elements are sorted, all occurrences of an element
        // must be together
        while (arr[i] == arr[i+1])
             i++;