How do permutations differ from combinations?

In combinatorics, the fundamental difference between permutations and combinations lies in whether the order of selection matters.1

• Permutations: A selection is considered a permutation when the order of the objects is a factor. For example, if you select two letters from the set {A, B, C, D, E}, the arrangements AB and BA are counted as two distinct permutations. The number of permutations of $n$ objects taken $k$ at a time is calculated as:1
  $${}_n{P}_k=\frac{n!}{(n-k)!}$$

• Combinations: A selection is considered a combination when the order of the objects is not a factor. In this case, AB and BA are treated as the same subset. The number of combinations of $n$ objects taken $k$ at a time is calculated by dividing the permutation formula by $k!$ to remove the redundant ordered arrangements:
  $${}_n{C}_k=\frac{n!}{k!(n-k)!}$$

Essentially, while permutations focus on the specific arrangement of elements, combinations focus solely on the membership of the resulting subset.1