Probability
Counting Arrangements – Rows, Words, and Repeated Items
Arranging n distinct items in a row is the simplest permutation case of all: there are exactly n! ways to do it. But when some of the items are identical – like the two “E”s in the word “GREEN” – swapping those identical items around doesn't create a new arrangement, so the raw n! count needs to be divided down.
The fix is simple: divide n! by the factorial of how many times each repeated item appears. This adjustment is exactly why word-arrangement problems ask you to check for repeated letters before applying the plain n! formula.
Arranging with Repeated Items
n distinct items in a row: n! arrangements. If one item repeats k times, divide by k! to remove the duplicate orderings.
How many distinct arrangements are there of the 5 letters in a word with all different letters?
5! = 5 × 4 × 3 × 2 × 1 = 120.
A 6-letter word has one letter repeated exactly twice. How many distinct arrangements are there?
6! ÷ 2! = 720 ÷ 2 = 360.
Real-Life Application
- Flag design: counting distinct arrangements of coloured stripes, some repeated.
- Music composition: counting distinct orderings of notes, some repeated.
- Anagram puzzles: counting the distinct anagrams of a word with repeated letters.
Key Takeaways
- n distinct items in a row can be arranged n! ways.
- Repeated items reduce the count, since swapping identical items doesn't create a new arrangement.
- Divide by k! for every item that repeats k times.
Practice: Counting Arrangements
Counting Arrangements