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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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 software 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 make service delivery on 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.

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

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;

}

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- How to generated 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 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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