Permutations and Combinations

Permutations vs. combinations

A permutation is an arrangement where order matters; a combinationis a selection where it doesn't. The number of ways to arrange all $n$ distinct items is $n!$ (n factorial): $n! = n \times (n-1) \times \cdots \times 2 \times 1$.

Worked example: the 19th word in dictionary order

If all permutations of the letters of the word MASK are arranged in dictionary order, which one is the 19th word?

Solution: Sort the letters alphabetically first: A, K, M, S. There are $4! = 24$ total arrangements. Group them by starting letter: each group has $3! = 6$ words, since the remaining 3 letters can be arranged 6 ways:

• Words 1–6 start with A
• Words 7–12 start with K
• Words 13–18 start with M
• Words 19–24 start with S

The 19th word is the first word starting with S. Arranging the remaining letters (A, K, M) in alphabetical order gives SAKM.