How Many Permutations Exist for a Given Set of Numbers, and How Can They Be Generated in PHP?

Finding All Possible Number Sets Using Permutations

Calculating all possible sets of numbers from a given range that use all numbers and allow each number to appear only once involves the mathematical concept of permutations. The permutation formula calculates the number of unique arrangements or orderings of a set of elements.

For a set of n numbers, where n! represents the factorial of n (n (n-1) (n-2) ... * 1), the total number of permutations is given by:

nPk = n!/(n-k)!

In this case, with 9 numbers and choosing all of them (k=n), the number of permutations becomes:

9P9 = 362,880

To generate these permutations in PHP, one can use this function provided by O'Reilly's "PHP Cookbook":

function pc_permute($items, $perms = array( )) {
    if (empty($items)) {
        print join(' ', $perms) . "\n";
    } else {
        for ($i = count($items) - 1; $i >= 0; --$i) {
            $newitems = $items;
            $newperms = $perms;
            list($foo) = array_splice($newitems, $i, 1);
            array_unshift($newperms, $foo);
            pc_permute($newitems, $newperms);
        }
    }
}

Calling this function with the set of numbers, such as:

pc_permute(array(0, 1, 2, 3, 4, 5, 7, 8));

will print out all possible permutations, including the examples provided:

0-1-2-3-4-5-6-7-8
0-1-2-3-4-5-6-8-7
0-1-2-3-4-5-8-6-7
0-1-2-3-4-8-5-6-7
0-1-2-3-8-4-5-6-7
0-1-2-8-3-4-5-6-7
...

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