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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⮞ Go to: Combinations with Repetition

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.)

In mathematics, a combination of k among n is the name given to a subset of k elements from another set consisting of n elements (with $ n \ge k $). In a combination, the order of the elements does not matter.

The generator allows selection of values $ k $ and $ n $, and generates possible lists 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 limit secures the computation server.

To generate larger 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 possible 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. The probability of winning is therefore 1 in 116 million.

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. The probability of winning is therefore 1 in 292 million.

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 0 choose 0 is 1: $$ \binom{0}{0} = 1 $$

`// 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));

}

// VBA

Function Factorial(n As Integer) As Double

Factorial = 1

For i = 1 To n

Factorial = Factorial * i

Next

End Function

Function NbCombinations (k As Integer, n As Integer) As Double

Dim z As Integer

z = n - k

NbCombinations = Factorial(n) / (Factorial(k) * Factorial(z))

End Function

`// 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;

}

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Cite as source (bibliography):

*Combination N Choose K* on dCode.fr [online website], retrieved on 2022-09-29,

- Combinations Generator
- Combinations with Order (1,2≠2,1)
- Combinations with Repeated Items
- Combinations Count Calculator
- Combinations and Lottery Games
- What is a combination of n choose k? (Definition)
- How to generate combinations of n choose k?
- How to count the number of combinations of n choose k?
- 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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