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Combination n choose k

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

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Combination n choose k -

Tag(s) : Mathematics,Combinatorics

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Combination n choose k

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Also on dCode: Permutations

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

Answers to Questions

How to generated combinations of n choose k?

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

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.

How to count the number of combinations of n choose k?

The calculation uses the binomial coefficient:$$ C_n^k = \binom{n}{k} = \frac{n!}{k!(n-k)!} $$Combinations uses calculus of factorialshref (the exclamation mark: !).

3 choose 23 combinations(1,2)(1,3)(2,3)
4 choose 26 combinations(1,2)(1,3)(1,4)(2,3)(2,4)(3,4)
5 choose 210 combinations(1,2)(1,3)(1,4)(1,5)(2,3)(2,4)(2,5)(3,4)(3,5)(4,5)
6 choose 215 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 221 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 228 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 236 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 34 combinations(1,2,3)(1,2,4)(1,3,4)(2,3,4)
5 choose 310 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 320 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 335 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 45 combinations(1,2,3,4)(1,2,3,5)(1,2,4,5)(1,3,4,5)(2,3,4,5)
6 choose 415 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 435 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 56 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 521 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)

How to take into account the order of the elements?

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

How many combinations is there to lottery/euromillions?

To win at EuroMillions, one draws 5 balls out of 50 (50 choose 5), then 2 stars out of 11 (11 choose 2).

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).

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.

Why k cannot be equal to zero 0?

If k = 0, then you ask for 0 item, there are no combination with 0 item.

Why n cannot be equal to zero 0?

If n is equal to zero, then there is no element to pick, the possible list is empty.

What is the algorithm for counting combinations?

// 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 factorialhref(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 factorialhref(x)/(factorialhref(y)*factorialhref(z));
}

What is the algorithm to generate combinations?

// 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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