The Counting Principle, Permutations, and Combinations

The number of ordered r-item selections from n distinct items without repetition is given by the formula $n!/(n-r)!$. This represents the product of the first r terms of the descending sequence starting from n.

Conditions: The items are distinct.; Selection is without repetition.; The order of selection matters.; $r$ is a non-negative integer less than or equal to $n$.

Dividing the permutation count by $r!$ yields the number of unordered groups because each specific set of $r$ items can be arranged in $r!$ different orders. Since combinations ignore internal order, all $r!$ permutations of a single group correspond to just one combination.

Conditions: The selection is without repetition.; The items are distinct.; We are converting from ordered selections (permutations) to unordered selections (combinations).

The multiplication principle determines the total number of outfits by multiplying the number of choices available at each independent stage. If there are multiple categories of items (like pants, shirts, and shoes) and you select exactly one item from each category, the total number of distinct combinations is the product of the counts of options in each category.

Conditions: The choices in each category are independent of the others.; Exactly one item is selected from each category.; The items within a category are distinct.