Why does dividing the permutation count by r! yield the number of unordered r-item groups?

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 $n$.

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.

For n distinct objects, successive choices are n, $n-1$, …, 1, giving n!. The counting rule needs the stated number of choices after each preceding choice; this is not a probability-independence assumption.

Conditions: Objects are distinct.; Order matters.