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.
|Replacements/Repetitions allowed||Replacements/Repetitions not allowed|
|Permutations/Order Important||nr possibilities
See the special case when r=n below
Here r=n, as we are permuting all the characters of the string.
|Combinations/Order Not Important||n+r-1Cr possibilities
Current Article ( https://www.geeksforgeeks.org/combinations-with-repetitions )
This article is about the third case(Order Not important and Repetitions allowed).
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.
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.
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 email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
- Make all combinations of size k
- Combinations in a String of Digits
- Combinations from n arrays picking one element from each array
- itertools.combinations() module in Python to print all possible combinations
- Problem on permutations and combinations | Set 2
- Print all the combinations of N elements by changing sign such that their sum is divisible by M
- Iterating over all possible combinations in an Array using Bits
- Iterative approach to print all combinations of an Array
- Program for nth Fuss–Catalan Number
- Count of subsets with sum equal to X
- Sum of subsets of all the subsets of an array | O(N)
- Count of pairs in an array such that the highest power of 2 that divides their product is 1
- Sum of Digits of the Good Strings
- Count non-adjacent subsets from numbers arranged in Circular fashion
Improved By : princiraj1992