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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