# combination

Returns the number of unique subsets created from all combinations of a number of elements. The order of the chosen elements is not significant.

$$\mathrm{C}_{k}^{n} = \frac{n\times(n-1)\times(n-2)\cdots(n-k+1)}{k!}$$

In a lottery, five numbers are selected from a pool of 49 unique numbers. The numbers are removed from the pool as they are selected and their order is not important. The number of possible outcomes is equal to \(C(49,5)\) or \(1906884\). The odds of correctly picking the one winning combination is \(1 / 1906884\).

In a card game, a player is dealt five cards from a standard deck of 52. What are the odds the player has been dealt a pair of aces? In order to be dealt exactly two aces, the player must be dealt a combination of two of four aces, \(C(4,2)\), and a combination of three of 48 non-ace cards, \(C(48,3)\), for a total of five cards. The odds are calculated by dividing the product of those combinations by the number of combinations of any five cards from a deck of 52. The odds of being dealt two aces equals \(C(4,2) \times C(48,3) \div C(52,5) = 0.0399298181\) or about 4%.

- combination(set, subset)
- Returns the number of unique subsets created from all combinations of a number of elements.

`COPY````
/// @func combination(set, subset)
///
/// @desc Returns the number of unique subsets created from all
/// combinations of a number of elements. The order of the
/// chosen elements is not significant. Returns (-1) on error.
///
/// @param {real} set number of elements
/// @param {real} subset size of the subset
///
/// @return {real} number of combinations
///
/// GMLscripts.com/license
function combination(set, subset)
{
var n = floor(set);
var k = floor(subset);
var m = n - k;
if (m < 0) return (-1);
var f = 1;
var i = 1;
var j = 1;
for (var l=2; l<=n; l+=1) {
f *= l;
if (l == k) i = f;
if (l == m) j = f;
}
return (f / (i * j));
}
```

Contributors: xot

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