Combination N Choose K

A combination of $ k $ among $ n $ is the name given to the number of distinct ways of choosing $ k $ elements among another set of $ n $ elements (with $ n \ge k $), without taking into account the order.

The combination is denoted by $ C_n^k $ or $ \binom{n}{k} $.

The generator allows selection of values $ k $ and $ n $, and generates possible lists of combinations with digits or letters (or a custom list).

The generation is limited to thousands of combinations. 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: !).

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 calculate the probabilities of winning in games of chance such as drawing random games, combinations are the most suitable tools.

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

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

To win at EuroDreams, the draw is 6 numbers from 40, then 1 number from 5.

Many books describes strategies for lotto or lottery such as here (affiliate 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;
}