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# Combinations with repetitions

Suppose we have a string of length- n and we want to generate all combinations/permutations taken r at a time with/without repetitions.  There are four fundamental concepts in Combinatorics
1) Combinations without repetitions/replacements.
2) Combinations with repetitions/replacements.
3) Permutations without repetitions/replacements.
4) Permutations with repetitions/replacements.
Below is a summary table depicting the fundamental concepts in Combinatorics Theory.

Summary Table

The idea is to recur for all the possibilities of the string, even if the characters are repeating.
The base case of the recursion is when there is a total of ‘r’ characters and the combination is ready to be printed.
For clarity, see the recursion tree for the string- “ 1 2 3 4” and r=2 Below is the implementation.

## C++

 `// C++ program to print all combination of size r in an array``// of size n with repetitions allowed``#include ``using` `namespace` `std;` `/* arr[]  ---> Input Array``  ``chosen[] ---> Temporary array to store indices of``                   ``current combination``   ``start & end ---> Starting and Ending indexes in arr[]``   ``r ---> Size of a combination to be printed */``void` `CombinationRepetitionUtil(``int` `chosen[], ``int` `arr[],``                    ``int` `index, ``int` `r, ``int` `start, ``int` `end)``{``    ``// Since index has become r, current combination is``    ``// ready to be printed, print``    ``if` `(index == r)``    ``{``        ``for` `(``int` `i = 0; i < r; i++)``            ``cout<<``" "``<< arr[chosen[i]];``        ``cout<<``"\n"``;``        ``return``;``    ``}` `    ``// One by one choose all elements (without considering``    ``// the fact whether element is already chosen or not)``    ``// and recur``    ``for` `(``int` `i = start; i <= end; i++)``    ``{``        ``chosen[index] = i;``        ``CombinationRepetitionUtil(chosen, arr, index + 1,``                                               ``r, i, end);``    ``}``    ``return``;``}` `// The main function that prints all combinations of size r``// in arr[] of size n with repetitions. This function mainly``// uses CombinationRepetitionUtil()``void` `CombinationRepetition(``int` `arr[], ``int` `n, ``int` `r)``{``    ``// Allocate memory``    ``int` `chosen[r+1];` `    ``// Call the recursive function``    ``CombinationRepetitionUtil(chosen, arr, 0, r, 0, n-1);``}` `// Driver program to test above functions``int` `main()``{``    ``int` `arr[] = {1, 2, 3, 4};``    ``int` `n = ``sizeof``(arr)/``sizeof``(arr);``    ``int` `r = 2;``    ``CombinationRepetition(arr, n, r);``    ``return` `0;``}``// this code is contributed by shivanisinghss2110`

## C

 `// C program to print all combination of size r in an array``// of size n with repetitions allowed``#include ` `/* arr[]  ---> Input Array``  ``chosen[] ---> Temporary array to store indices of``                   ``current combination``   ``start & end ---> Starting and Ending indexes in arr[]``   ``r ---> Size of a combination to be printed */``void` `CombinationRepetitionUtil(``int` `chosen[], ``int` `arr[],``                    ``int` `index, ``int` `r, ``int` `start, ``int` `end)``{``    ``// Since index has become r, current combination is``    ``// ready to be printed, print``    ``if` `(index == r)``    ``{``        ``for` `(``int` `i = 0; i < r; i++)``            ``printf``(``"%d "``, arr[chosen[i]]);``        ``printf``(``"\n"``);``        ``return``;``    ``}` `    ``// One by one choose all elements (without considering``    ``// the fact whether element is already chosen or not)``    ``// and recur``    ``for` `(``int` `i = start; i <= end; i++)``    ``{``        ``chosen[index] = i;``        ``CombinationRepetitionUtil(chosen, arr, index + 1,``                                               ``r, i, end);``    ``}``    ``return``;``}` `// The main function that prints all combinations of size r``// in arr[] of size n with repetitions. This function mainly``// uses CombinationRepetitionUtil()``void` `CombinationRepetition(``int` `arr[], ``int` `n, ``int` `r)``{``    ``// Allocate memory``    ``int` `chosen[r+1];` `    ``// Call the recursive function``    ``CombinationRepetitionUtil(chosen, arr, 0, r, 0, n-1);``}` `// Driver program to test above functions``int` `main()``{``    ``int` `arr[] = {1, 2, 3, 4};``    ``int` `n = ``sizeof``(arr)/``sizeof``(arr);``    ``int` `r = 2;``    ``CombinationRepetition(arr, n, r);``    ``return` `0;``}`

## Java

 `// Java program to print all combination of size r in an array``// of size n with repetitions allowed` `class` `GFG {` `    ``/* arr[] ---> Input Array``chosen[] ---> Temporary array to store indices of``                ``current combination``start & end ---> Starting and Ending indexes in arr[]``r ---> Size of a combination to be printed */``    ``static` `void` `CombinationRepetitionUtil(``int` `chosen[], ``int` `arr[],``            ``int` `index, ``int` `r, ``int` `start, ``int` `end) {``        ``// Since index has become r, current combination is``        ``// ready to be printed, print``        ``if` `(index == r) {``            ``for` `(``int` `i = ``0``; i < r; i++) {``                ``System.out.printf(``"%d "``, arr[chosen[i]]);``            ``}``            ``System.out.printf(``"\n"``);``            ``return``;``        ``}` `        ``// One by one choose all elements (without considering``        ``// the fact whether element is already chosen or not)``        ``// and recur``        ``for` `(``int` `i = start; i <= end; i++) {``            ``chosen[index] = i;``            ``CombinationRepetitionUtil(chosen, arr, index + ``1``,``                    ``r, i, end);``        ``}``        ``return``;``    ``}` `// The main function that prints all combinations of size r``// in arr[] of size n with repetitions. This function mainly``// uses CombinationRepetitionUtil()``    ``static` `void` `CombinationRepetition(``int` `arr[], ``int` `n, ``int` `r) {``        ``// Allocate memory``        ``int` `chosen[] = ``new` `int``[r + ``1``];` `        ``// Call the recursive function``        ``CombinationRepetitionUtil(chosen, arr, ``0``, r, ``0``, n - ``1``);``    ``}` `// Driver program to test above functions``    ``public` `static` `void` `main(String[] args) {``        ``int` `arr[] = {``1``, ``2``, ``3``, ``4``};``        ``int` `n = arr.length;``        ``int` `r = ``2``;``        ``CombinationRepetition(arr, n, r);``    ``}``}` `/* This Java code is contributed by PrinciRaj1992*/`

## Python3

 `# Python3 program to print all combination``# of size r in an array of size n` `''' arr[] ---> Input Array``    ``chosen[] ---> Temporary array to store``               ``current combination``    ``start & end ---> Starting and Ending indexes in arr[]``    ``r---> Size of a combination to be printed` `    ``'''``def` `CombinationRepetitionUtil(chosen, arr, index,``                              ``r, start, end):``                                  ` `    ``# Current combination is ready,``    ``# print it``    ``if` `index ``=``=` `r:``        ``for` `j ``in` `range``(r):``            ``print``(chosen[j], end ``=` `" "``)``            ` `        ``print``()``        ``return``        ` `    ``# When no more elements are``    ``# there to put in chosen[]``    ``if` `start > n:``        ``return``        ` `    ``# Current is included, put``    ``# next at next location``    ``chosen[index] ``=` `arr[start]``    ` `    ``# Current is excluded, replace it``    ``# with next (Note that i+1 is passed,``    ``# but index is not changed)``    ``CombinationRepetitionUtil(chosen, arr, index ``+` `1``,``                              ``r, start, end)``    ``CombinationRepetitionUtil(chosen, arr, index,``                              ``r, start ``+` `1``, end)` `# The main function that prints all``# combinations of size r in arr[] of``# size n. This function mainly uses``# CombinationRepetitionUtil()``def` `CombinationRepetition(arr, n, r):``    ` `    ``# A temporary array to store``    ``# all combination one by one``    ``chosen ``=` `[``0``] ``*` `r` `    ``# Print all combination using``    ``# temporary array 'chosen[]'``    ``CombinationRepetitionUtil(chosen, arr, ``0``, r, ``0``, n)` `# Driver code``arr ``=` `[ ``1``, ``2``, ``3``, ``4` `]``r ``=` `2``n ``=` `len``(arr) ``-` `1` `CombinationRepetition(arr, n, r)` `# This code is contributed by Vaibhav Kumar 12.`

## C#

 `// C# program to print all combination of size r in an array``// of size n with repetitions allowed` `using` `System;``public` `class` `GFG{`  `    ``/* arr[] ---> Input Array``chosen[] ---> Temporary array to store indices of``                ``current combination``start & end ---> Starting and Ending indexes in arr[]``r ---> Size of a combination to be printed */``    ``static` `void` `CombinationRepetitionUtil(``int` `[]chosen, ``int` `[]arr,``            ``int` `index, ``int` `r, ``int` `start, ``int` `end) {``        ``// Since index has become r, current combination is``        ``// ready to be printed, print``        ``if` `(index == r) {``            ``for` `(``int` `i = 0; i < r; i++) {``                ``Console.Write(arr[chosen[i]]+``" "``);``            ``}``            ``Console.WriteLine();``            ``return``;``        ``}` `        ``// One by one choose all elements (without considering``        ``// the fact whether element is already chosen or not)``        ``// and recur``        ``for` `(``int` `i = start; i <= end; i++) {``            ``chosen[index] = i;``            ``CombinationRepetitionUtil(chosen, arr, index + 1,``                    ``r, i, end);``        ``}``        ``return``;``    ``}` `// The main function that prints all combinations of size r``// in arr[] of size n with repetitions. This function mainly``// uses CombinationRepetitionUtil()``    ``static` `void` `CombinationRepetition(``int` `[]arr, ``int` `n, ``int` `r) {``        ``// Allocate memory``        ``int` `[]chosen = ``new` `int``[r + 1];` `        ``// Call the recursive function``        ``CombinationRepetitionUtil(chosen, arr, 0, r, 0, n - 1);``    ``}` `// Driver program to test above functions``    ``public` `static` `void` `Main() {``        ``int` `[]arr = {1, 2, 3, 4};``        ``int` `n = arr.Length;``        ``int` `r = 2;``        ``CombinationRepetition(arr, n, r);``    ``}``}` `// This code is contributed by PrinciRaj1992`

## Javascript

 ``

Output :

```1 1
1 2
1 3
1 4
2 2
2 3
2 4
3 3
3 4
4 4```

Time Complexity:  For a string of length- n and combinations taken r at a time with repetitions, it takes a total of O(n+r-1Cr) time.
Referenceshttps://en.wikipedia.org/wiki/Combination
This article is contributed by Rachit Belwariar. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.