What distinguishes permutations from combinations?

Permutations are arrangements of items where the order of the items matters. For example, the arrangements of the letters in the word "ABC" would be "ABC," "ACB," "BAC," "BCA," "CAB," and "CBA," making six distinct arrangements. This illustrates that changing the order of the items creates a different arrangement.

In contrast, combinations refer to selections of items where the order does not matter. For instance, the combination of letters "A," "B," and "C" would be the same regardless of the order in which they are chosen; "A," "B," and "C" would be considered the same combination as "B," "A," and "C."

Thus, the key distinction lies in the dependency on order: permutations require a specific arrangement, while combinations involve selection without regard to order. Understanding this difference is critical in both combinatorial mathematics and practical applications in areas like probability and statistics.