**Wong Shao Voon**on

**August 4th, 2003**

### Introduction

A *combination* is the way of picking a different unique smaller set from a bigger set, without regard to the ordering (positions) of the elements (in the smaller set). This article teaches you how to find combinations. First, I show you the technique to find combinations. Next, I will go on to explain how to use my source code. The source includes a recursive template version and a non-recursive template version. At the end of article, I will show you how to find permutations of a smaller set from a bigger set, using both `next_combination()` and `next_permutation()`.

Before all these, let me first introduce to you the technique of finding combinations.

### The Technique

**n**: The larger sequence from which r sequence is picked**r**: The smaller sequence picked from n sequence**c**: The formula for the total number of possible combinations of r picked from n distinct objects: n! / (r! (n-r)! )

Note:The ! postfix means factorial.

### Explanation

Let me explain using a very simple example: finding all combinations of 2 from a set of 6 letters {A, B, C, D, E, F}. The first combination is AB and the last is EF.

The total number of possible combinations is: n!/(r!(n-r)!)=6!/(2!(6-2)!)=15 combinations.

Let me show you all the combinations first:

AB AC AD AE AF BC BD BE BF CD CE CF DE DF EF

If you can't spot the pattern, here it is:

AB | AB A | AC A | AD A | AE A | AF ---|---- BC | BC B | BD B | BE B | BF ---|---- CD | CD C | CE C | CF ---|---- DE | DE D | DF ---|---- EF | EF

The same thing goes for combinations of any numbers of letters. Let me give you few more examples and now you can figure them out yourself.

#### Combinations of 3 letters from {A, B, C, D, E} (set of 5 letters)

The total number of possible combinations is: 10

A B C A B D A B E A C D A C E A D E B C D B C E B D E C D E

#### Combinations of 4 letters from {A, B, C, D, E, F} (set of 6 letters)

The total number of possible combinations is: 15

A B C D A B C E A B C F A B D E A B D F A B E F A C D E A C D F A C E F A D E F B C D E B C D F B C E F B D E F C D E F

I'm thinking that you would have noticed by now the number of times that a letter appears. The formula for the number of times a letter appears in all possible combinations is n!/(r!(n-r)!) * r / n == c * r / n. Using the above example, it would be 15 * 4 / 6 = 10 times. All the letters {A, B, C, D, E, F} appear 10 times as shown. You can count them yourself to prove it.

Now, go on to the source code section.

### Source Code Section

Please note that all the combination functions are now enclosed in the stdcomb namespace.

#### The recursive way

I have made a recursive function, `char_combination()`, which, as its name implies, takes in character arrays and processes them. The source code and examples of using `char_combination()` is in `char_comb_ex.cpp`. I'll stop to mention that function. For now, your focus is on `recursive_combination()`, a template function that I wrote, using `char_combination()` as a guideline.

The function defined in combination.h as below:

// Recursive template function template <class RanIt, class Func> void recursive_combination(RanIt nbegin, RanIt nend, int n_column, RanIt rbegin, RanIt rend, int r_column,int loop, Func func) { int r_size=rend-rbegin; int localloop=loop; int local_n_column=n_column; //A different combination is out if(r_column>(r_size-1)) { func(rbegin,rend); return; } //=========================== for(int i=0;i<=loop;++i) { RanIt it1=rbegin; for(int cnt=0;cnt<r_column;++cnt) { ++it1; } RanIt it2=nbegin; for(int cnt2=0;cnt2<n_column+i;++cnt2) { ++it2; } *it1=*it2; ++local_n_column; recursive_combination(nbegin,nend,local_n_column, rbegin,rend,r_column+1,localloop,func); --localloop; } }

The parameters prefixed with 'n' are associated with n sequence, whereas r-prefix parameters are r sequence related. As a end user, you need not bother about those parameters. What you need to know is `func`. `func` is a function that you defined. If the combination function finds a combination recursively, a way the user can process each combination must exist. The solution is a function pointer that takes in two parameters of type `RanIt` (stands for Random Iterator). You are the one who defines this function. In this way, encapsulation is achieved. You need not know how `recursive_combination()` internally works; you just know that it calls `func` whenever there is a different combination, and you just need to define the `func()` function to process the combination.

Note:func()should not write to the two iterators passed to it.

The typical way of filling out the parameters in `n_column` and `r_column` is always 0, `loop` is the number of elements in r sequence minus that of n sequence, `func` is the function pointer to your function. (`nbegin` and `nend`, `rbegin` and `rend` are self-explanatory; they are the first iterators and the one past the last iterators, of the respective sequences.)

Just for your information, the maximum number of depth of recursion done is r+1. In the last recursion (r+1 recursion), each new combination is formed.

An example of using recursive_combination() with raw character arrays is below

#include <iostream> #include <vector> #include <string> #include "combination.h" using namespace std; using namespace stdcomb; void display(char* begin,char* end) { cout<<begin<<endl; } int main() { char ca[]="123456"; char cb[]="1234"; recursive_combination(ca,ca+6,0, cb,cb+4,0,6-4,display); cout<<"Complete!"<<endl; return 0; }

An example of using recursive_combination() with vector of integers is below

#include <iostream> #include <vector> #include <string> #include "combination.h" typedef vector::iterator vii; void display(vii begin,vii end) { for (vii it=begin;it!=end;++it) cout<<*it; cout<<endl; } int main() { vector<int> ca; ca.push_back (1); ca.push_back (2); ca.push_back (3); ca.push_back (4); ca.push_back (5); ca.push_back (6); vector<int> cb; cb.push_back (1); cb.push_back (2); cb.push_back (3); cb.push_back (4); recursive_combination(ca.begin (),ca.end(),0, cb.begin(),cb.end(),0,6-4,display); cout<<"Complete!"<<endl; return 0; }

### The Non-Recursive Way

If you have misgivings about using the recursive method, there is a non-recursive template function for you to choose. (Actually, there are two.)

The parameters are even simpler than the recursive version. Here's the function definition in `combination.h`.

template <class BidIt> bool next_combination(BidIt n_begin, BidIt n_end, BidIt r_begin, BidIt r_end); template <class BidIt> bool next_combination(BidIt n_begin, BidIt n_end, BidIt r_begin, BidIt r_end, Prediate Equal );

And its reverse counterpart version,

template <class BidIt> bool prev_combination(BidIt n_begin, BidIt n_end, BidIt r_begin, BidIt r_end); template <class BidIt> bool prev_combination(BidIt n_begin, BidIt n_end, BidIt r_begin, BidIt r_end, , Prediate Equal );

The parameters `n_begin` and `n_end` are the first and last iterators for the n sequence. And, `r_begin` and `r_end` are iterators for the r sequence. Equal is the prediate for comparing equality.

You can peruse the source code for these two functions in `combination.h` and its examples in `next_comb_ex.cpp` and `prev_comb_ex.cpp` if you want.

A typical way of using next_combination with raw character arrays is as below:

#include <iostream> #include <vector> #include <string> #include "combination.h" using namespace std; using namespace stdcomb; int main() { char ca[]="123456"; char cb[]="1234"; do { cout<<cb<<endl; } while(next_combination(ca,ca+6,cb,cb+4)); cout<<"Complete!"<<endl; return 0; }

A typical way of using next_combination with vector of integers is as below:

#include <iostream> #include <vector> #include <string> #include "combination.h" template<class BidIt> void display(BidIt begin,BidIt end) { for (BidIt it=begin;it!=end;++it) cout<<*it<<" "; cout<<endl; } int main() { vector<int> ca; ca.push_back (1); ca.push_back (2); ca.push_back (3); ca.push_back (4); ca.push_back (5); ca.push_back (6); vector<int> cb; cb.push_back (1); cb.push_back (2); cb.push_back (3); cb.push_back (4); do { display(cb.begin(),cb.end()); } while(next_combination(ca.begin (),ca.end (),cb.begin (),cb.end()) ); cout<<"Complete!"<<endl; return 0; }

Certain conditions must be satisfied in order for next_combination() to work:

- All the objects in the n sequence must be distinct.
- For
`next_combination()`, the r sequence must be initialised to the first r-th elements of n sequence in the first call. For example, to find combinations of r=4 out of n=6 {1,2,3,4,5,6}, the r sequence must be initialsed to {1,2,3,4} before the first call. - As for
`prev_combination()`, the r sequence must be initialised to the last r-th elements of n sequence in the first call. For example, to find combinations of r=4 out of n=6 {1,2,3,4,5,6}, the r sequence must be initialsed to {3,4,5,6} before the first call. - The n sequence must not change thoughout the process of finding all the combinations; otherwise, results are wrong (makes sense, right?).
`next_combination()`and`prev_combination()`operate on data types with the`==`operator defined. This means that if you want to use`next_combination()`on sequences of objects instead of sequences of POD (Plain Old Data), the class from these objects are instantiated must have an overloaded`==`operator defined or you can use the prediate versions.

When the above conditions are not satisfied, results are undetermined even if the `next_combination()` and `prev_combination()` may return true.

### Return Value

When `next_combination()` returns false, no more next combination can be found; the r sequence remains unaltered. The same is true for `prev_combination()`.

### Some Information about next_combination() and prev_combination()

- The n and r sequences need not be sorted to use
`next_combination()`or`prev_combination()`. `next_combination()`and`prev_combination()`do not use any static variables, so it is all right to find combinations of another sequence of a different data type, even when the current finding of combinations of the current sequence has not reached the last combination. In other words, no reset is needed for`next_combination()`and`prev_combination()`.

Examples of how to use these two functions are in `next_comb_ex.cpp` and `prev_comb_ex.cpp`.

### What You Can Do with next_combination()

With `next_combination()` and `next_permutation()` from the STL algorithms, you can find permutations!! The formula for total number of permutations of r sequence picked from n sequence is n!/(n-r)!

You can call `next_combination()` first and then `next_permutation()` iteratively. That way, you will find all the permutations. A typical way of using them is as follows:

sort(n.begin(),n.end()); do { sort(r.begin(),r.end()); //do your processing on the new combination here do { //do your processing on the new permutation here } while(next_permutation(r2.begin(),r2.end())) } while(next_combination(n.begin(),n.end(),r.begin(),r.end() ));

However, I must mention a limitation for the above code exists. The n and r sequences must be sorted in ascending order to work. This is because `next_permutation()` will return false when it encounters the sequence in descending order. The solution to this problem for unsorted sequences is as follows:

do { //do your processing on the new combination here for(cnt i=0;cnt<24;++cnt) { next_permutation(r2.begin(),r2.end()); //do your processing on the new permutation here } } while(next_combination(n.begin(),n.end(),r.begin(),r.end() ));

However, this method requires you to calculate the number of permutations beforehand.

### How to Prove They Are Distinct Permutations

You can use a `set` container class in STL. All the objects in the `set` container are always in sorted order and there are no duplicate objects. For uour purpose, you will use this `insert()` member function:

pair <iterator, bool> insert(const value_type& _Val);

The `insert()` member function returns a pair whose `bool` component returns `true` if an insertion is made and false if the `set` already contains an element whose key had an equivalent value in the ordering, and whose iterator component returns the address where a new element is inserted or where the element is already located.

proof.cpp is written for this purpose, using an STL set container to prove that the permutations generated are unique. You can play around with this, but you should first calculate the number of permutations that would be generated. Too many permutations may take ages to complete (partly due to working of the set container) or worse, you may run out of memory!

**Note:** I have written a Combinations in C++, Part 2 article which you may read if you are interested to find out more about computing combinations on multi-core machines and computing combinations with repeated elements.

### Revision History

14 September 2009 - Added the example code

17 March 2008 - Added the finding combinations of vectors in the source code

26 November 2006: Source code changes and bugs fixed:

- All functions are enclosed in stdcomb namespace.
- Solved a bug in prev_combination that != operator must be defined for the custom class, unless the data type is a POD.
- next_combination and prev_combination now runs properly in Visual C++ 8.0, without disabling the checked iterator.
- next_combination and prev_combination have prediates version.

## Comments

## Another sweet problem

Posted bySimonon07/09/2012 01:04pmI want to find all possible combination of letters and dot â.â Condition is that suppose if we have 2 letter "ab" then using "." in between them we can make combination as âa.bâ Same for 3 letters "abc" we can make possible combination "a.bc", "ab.c" and "a.b.c" Same for 4 letter "abcd" we can make "a.bcd", "ab.cd", "abc.d", "a.b.cd", "a.b.c.d", "ab.cd", "ab.c.d", "a.bc.d" and "abc.d". For the above problem I have tried to make a c++ program to find all possible combinations for any number of letters but I haven't been able to find a good solution.

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Posted byCBasicNeton12/04/2006 04:42amHi guys, do not read this article because it is not the updated one, it is still the old one. I'll report this to Codeguru webmaster.

## Article is updated.

Posted byCBasicNeton12/05/2006 04:39amBrad has resolved it for me. Thanks, Brad!

Reply## Combinations in C++ using numbers as input

Posted byComputerPublicon11/25/2006 12:48amHi, I have used your combination for char, but when I tried changing it to integers, it did not work. Can you please show me how to make it work for numbers. Thanks. Ed.

Reply## Bit Combinations

Posted byLegacyon09/11/2003 07:00amOriginally posted by: Jimmy

Hi, using your algorithms, I've generate combinations for the integers 1..20 : this gives over 1 million combos : thanks for that.

ReplyMy program needs to filter out some of these combinations eg. don't allow combinations which contain {2,3). No problem there but I want to increase the efficiency of my program both for memory usage and speed. I'm thinking of representing each combination as a bitset and using shift operations for comparisons and filtering out various combinations.

Is there come way I can use your algoritm to come up with all the combinations of 1's and 0's for say 20 bits eg

00000000000000000001 : 1

00000000000000000011 : 1,2

00000000000000000010 : 2

00000000000000000111 : 1,2,3

00000000000000000110 : 2,3

00000000000000000101 : 1,3

....

## what about,,,

Posted byLegacyon09/01/2003 07:00amOriginally posted by: jared

what about reverse combos, ie BA, CB, etc.

what about redundancies, ie AA, BB, CC, etc, in the case of unlimited pool.

what about different variants on the letter, ie upper case, lower case, italic, bold, etc.

Is there a general formula for this?

## Another method using array of bools

Posted byLegacyon07/01/2003 07:00amOriginally posted by: Paul McKenzie

I've posted another method that overcomes

a) the problems of next_permutation returning false

b) is non-recursive

c) works for any type of data (since it is a template).

In short, it uses a totally different approach in finding combinations using next_permutation.

It is available here:

http://www.codeguru.com/forum/attachment.php?s=&postid=721348

The discussion is here:

http://www.codeguru.com/forum/showthread.php?s=&threadid=241675

Probably I will post an article on it in the near future.

Paul McKenzie

Reply## another algorithm for combinations

Posted byLegacyon05/27/2003 07:00amOriginally posted by: raffael cavaliere

I find it not good at all to use recursion when your "n" ie number of item in the collection is greater than 12 or 13. Stack errors are very easy to get when it occurs.

I wrote a small method you can use this way :

//JAVA

int[] list = {0,1,2,3,4,5}; //first elements (set of 6)

int n = 12; //number of items in the collection

while {list != null)

list = Clist(n, list);

public final static int[] CList(int n, int a[])

{

for (int i = a.length - 1; i >= 0; i--)

{

if (a[i] < n - a.length + i)

{

a[i]++;

for (int r = i + 1; r < a.length; r++)

a[r] = a[r - 1] + 1;

return a;

}

}

return null;

}

## Another point to mention!

Posted byLegacyon02/19/2003 08:00amOriginally posted by:ButWhat if the location of the letter is important. I.e ABC!=CBA.

AAA

AAB

AAC

ABA

ABB

ABC

ACA

ACB

ACC

BAA

BAB

BAC

BBA

BBB

BBC

BCA

BCB

BCC

CAA

CAB

CAC

CBA

CBB

CBC

CCA

CCB

CCC

Total = 27

So if we can use n letters {A,B,C} and r rows then n^r

Reply## Combinations Tutorial

Posted byLegacyon02/18/2003 08:00amOriginally posted by: Big Al

Hi Thanks for your effort, could you explain how to do the same thing in VB?

Thanks

ReplyAl van Heerden

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