Tool to generate combinations. In mathematics, a choice of k elements out of n distinguishable objects (k choose n), where the order does not matter, is represented by a list of elements, which cardinal is the binomial coefficient.

Combination N Choose K - dCode

Tag(s) : Combinatorics

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To get a list of combinations with a guaranteed minimum of numbers (also called reduced lottery draw), dCode has a tool for that:

Go to: Covering Design for Lottery

To draw random numbers (Lotto, Euromillions, Superlotto, etc.)

Tool to generate combinations. In mathematics, a choice of k elements out of n distinguishable objects (k choose n), where the order does not matter, is represented by a list of elements, which cardinal is the binomial coefficient.

The generator allows to select values of $ k $ and $ n $, and generates list of **combinations** with digits or letters (or a custom list).

__Example:__ 4 choose 2 generates: (1,2),(1,3),(1,4),(2,3),(2,4),(3,4)

The generation is limited to 2000 lines. Combinatorics can introduce huge numbers, this limits secures the computation server.

To generates large lists, dCode can generate them upon (paid) request.

The calculation uses the binomial coefficient: $$ C_n^k = \binom{n}{k} = \frac{n!}{k!(n-k)!} $$

**Combinations** uses calculus of factorials (the exclamation mark: !).

3 choose 2 = 3 combinations | (1,2)(1,3)(2,3) |

4 choose 2 = 6 combinations | (1,2)(1,3)(1,4)(2,3)(2,4)(3,4) |

5 choose 2 = 10 combinations | (1,2)(1,3)(1,4)(1,5)(2,3)(2,4)(2,5)(3,4)(3,5)(4,5) |

6 choose 2 = 15 combinations | (1,2)(1,3)(1,4)(1,5)(1,6)(2,3)(2,4)(2,5)(2,6)(3,4)(3,5)(3,6)(4,5)(4,6)(5,6) |

7 choose 2 = 21 combinations | (1,2)(1,3)(1,4)(1,5)(1,6)(1,7)(2,3)(2,4)(2,5)(2,6)(2,7)(3,4)(3,5)(3,6)(3,7)(4,5)(4,6)(4,7)(5,6)(5,7)(6,7) |

8 choose 2 = 28 combinations | (1,2)(1,3)(1,4)(1,5)(1,6)(1,7)(1,8)(2,3)(2,4)(2,5)(2,6)(2,7)(2,8)(3,4)(3,5)(3,6)(3,7)(3,8)(4,5)(4,6)(4,7)(4,8)(5,6)(5,7)(5,8)(6,7)(6,8)(7,8) |

9 choose 2 = 36 combinations | (1,2)(1,3)(1,4)(1,5)(1,6)(1,7)(1,8)(1,9)(2,3)(2,4)(2,5)(2,6)(2,7)(2,8)(2,9)(3,4)(3,5)(3,6)(3,7)(3,8)(3,9)(4,5)(4,6)(4,7)(4,8)(4,9)(5,6)(5,7)(5,8)(5,9)(6,7)(6,8)(6,9)(7,8)(7,9)(8,9) |

4 choose 3 = 4 combinations | (1,2,3)(1,2,4)(1,3,4)(2,3,4) |

5 choose 3 = 10 combinations | (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) |

6 choose 3 = 20 combinations | (1,2,3)(1,2,4)(1,2,5)(1,2,6)(1,3,4)(1,3,5)(1,3,6)(1,4,5)(1,4,6)(1,5,6)(2,3,4)(2,3,5)(2,3,6)(2,4,5)(2,4,6)(2,5,6)(3,4,5)(3,4,6)(3,5,6)(4,5,6) |

7 choose 3 = 35 combinations | (1,2,3)(1,2,4)(1,2,5)(1,2,6)(1,2,7)(1,3,4)(1,3,5)(1,3,6)(1,3,7)(1,4,5)(1,4,6)(1,4,7)(1,5,6)(1,5,7)(1,6,7)(2,3,4)(2,3,5)(2,3,6)(2,3,7)(2,4,5)(2,4,6)(2,4,7)(2,5,6)(2,5,7)(2,6,7)(3,4,5)(3,4,6)(3,4,7)(3,5,6)(3,5,7)(3,6,7)(4,5,6)(4,5,7)(4,6,7)(5,6,7) |

5 choose 4 = 5 combinations | (1,2,3,4)(1,2,3,5)(1,2,4,5)(1,3,4,5)(2,3,4,5) |

6 choose 4 = 15 combinations | (1,2,3,4)(1,2,3,5)(1,2,3,6)(1,2,4,5)(1,2,4,6)(1,2,5,6)(1,3,4,5)(1,3,4,6)(1,3,5,6)(1,4,5,6)(2,3,4,5)(2,3,4,6)(2,3,5,6)(2,4,5,6)(3,4,5,6) |

7 choose 4 = 35 combinations | (1,2,3,4)(1,2,3,5)(1,2,3,6)(1,2,3,7)(1,2,4,5)(1,2,4,6)(1,2,4,7)(1,2,5,6)(1,2,5,7)(1,2,6,7)(1,3,4,5)(1,3,4,6)(1,3,4,7)(1,3,5,6)(1,3,5,7)(1,3,6,7)(1,4,5,6)(1,4,5,7)(1,4,6,7)(1,5,6,7)(2,3,4,5)(2,3,4,6)(2,3,4,7)(2,3,5,6)(2,3,5,7)(2,3,6,7)(2,4,5,6)(2,4,5,7)(2,4,6,7)(2,5,6,7)(3,4,5,6)(3,4,5,7)(3,4,6,7)(3,5,6,7)(4,5,6,7) |

6 choose 5 = 6 combinations | (1,2,3,4,5)(1,2,3,4,6)(1,2,3,5,6)(1,2,4,5,6)(1,3,4,5,6)(2,3,4,5,6) |

7 choose 5 = 21 combinations | (1,2,3,4,5)(1,2,3,4,6)(1,2,3,4,7)(1,2,3,5,6)(1,2,3,5,7)(1,2,3,6,7)(1,2,4,5,6)(1,2,4,5,7)(1,2,4,6,7)(1,2,5,6,7)(1,3,4,5,6)(1,3,4,5,7)(1,3,4,6,7)(1,3,5,6,7)(1,4,5,6,7)(2,3,4,5,6)(2,3,4,5,7)(2,3,4,6,7)(2,3,5,6,7)(2,4,5,6,7)(3,4,5,6,7) |

By principle, **combinations** do not take into account order (1,2) = (2,1). Use the function permutations to get ordered **combinations**.

dCode has a dedicated tool for **combinations** with repetitions.

To win at EuroMillions, a player ticks 5 boxes out of 50 (50 choose 5), then 2 stars out of 11 (11 choose 2).

__Example:__ Calculate the number of **combinations** of (50 choose 5) = 2 118 760, and multiply by (11 choose 2) = 55 for a total of 116 531 800 **combinations**.

To win at Powerball, pick 5 out of 69 (69 choose 5), then pick 1 out of 26 (26 choose 1).

__Example:__ Calculate the number of **combinations** of (69 choose 5) = 11 238 513, and multiply by (26 choose 1) = 26 for a total of 292 201 338 **combinations**.

Many books describes strategies for lotto or lottery such as here (link) One of the strategies is to play covering designs systems.

If $ k = 0 $, then 0 item are wanted, there is an empty result with 0 item. So $$ \binom{n}{0} = 1 $$

If $ n = 0 $, then there is 0 item, impossible to pick $ k $, so there are no results. So $$ \binom{0}{k} = 0 $$

By convention $$ \binom{0}{0} = 1 $$

`// pseudo code`

start count_combinations( k , n ) {

if (k = n) return 1;

if (k > n/2) k = n-k;

res = n-k+1;

for i = 2 by 1 while i < = k

res = res * (n-k+i)/i;

end for

return res;

end`// language C`

double factorial(double x) {

double i;

double result=1;

if (x >= 0) {

for(i=x;i>1;i--) {

result = result*i;

}

return result;

}

return 0; // error

}

double count_combinations(double x,double y) {

double z = x-y;

return factorial(x)/(factorial(y)*factorial(z));

}

`// javascript`

function **combinations**(a) { // a = new Array(1,2)

var fn = function(n, src, got, all) {

if (n == 0) {

if (got.length > 0) {

all[all.length] = got;

}

return;

}

for (var j = 0; j < src.length; j++) {

fn(n - 1, src.slice(j + 1), got.concat([src[j]]), all);

}

return;

}

var all = [];

for (var i=0; i < a.length; i++) {

fn(i, a, [], all);

}

all.push(a);

return all;

}

dCode retains ownership of the source code of the script Combination N Choose K online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Combination N Choose K script for offline use on PC, iPhone or Android, ask for price quote on contact page !

- Combinations Generator
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- How to take into account the order of the elements?
- How to get combinations with repetitions?
- How many combinations is there to lottery/euromillions?
- Why k cannot be equal to zero 0?
- Why n cannot be equal to zero 0?
- What is the value of 0 choose 0?
- What is the algorithm for counting combinations?
- What is the algorithm to generate combinations?

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Source : https://www.dcode.fr/combinations

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